MAYBE Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(U11(tt(),V1)) -> isNatList#(V1) active#(U11(tt(),V1)) -> U12#(isNatList(V1)) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> isNatList#(V) active#(U31(tt(),V)) -> U32#(isNatList(V)) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt(),V1,V2)) -> isNat#(V1) active#(U41(tt(),V1,V2)) -> U42#(isNat(V1),V2) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) active#(U42(tt(),V2)) -> isNatIList#(V2) active#(U42(tt(),V2)) -> U43#(isNatIList(V2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) active#(U43(tt())) -> mark#(tt()) active#(U51(tt(),V1,V2)) -> isNat#(V1) active#(U51(tt(),V1,V2)) -> U52#(isNat(V1),V2) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> isNatList#(V2) active#(U52(tt(),V2)) -> U53#(isNatList(V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) active#(U53(tt())) -> mark#(tt()) active#(U61(tt(),V1,V2)) -> isNat#(V1) active#(U61(tt(),V1,V2)) -> U62#(isNat(V1),V2) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) active#(U62(tt(),V2)) -> isNatIList#(V2) active#(U62(tt(),V2)) -> U63#(isNatIList(V2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) active#(U63(tt())) -> mark#(tt()) active#(U71(tt(),L)) -> length#(L) active#(U71(tt(),L)) -> s#(length(L)) active#(U71(tt(),L)) -> mark#(s(length(L))) active#(U81(tt())) -> mark#(nil()) active#(U91(tt(),IL,M,N)) -> take#(M,IL) active#(U91(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(length(V1))) -> isNatIListKind#(V1) active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatIList(V)) -> isNatIListKind#(V) active#(isNatIList(V)) -> U31#(isNatIListKind(V),V) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIList(cons(V1,V2))) -> U41#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(take(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(length(V1))) -> isNatIListKind#(V1) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(cons(V1,V2))) -> U51#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(take(V1,V2))) -> isNatKind#(V1) active#(isNatList(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(take(V1,V2))) -> U61#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> isNatKind#(N) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatKind(N)) active#(length(cons(N,L))) -> isNatIListKind#(L) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNatIListKind(L)) active#(length(cons(N,L))) -> and#(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) active#(length(cons(N,L))) -> U71#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active#(take(0(),IL)) -> isNatIListKind#(IL) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(0(),IL)) -> U81#(and(isNatIList(IL),isNatIListKind(IL))) active#(take(0(),IL)) -> mark#(U81(and(isNatIList(IL),isNatIListKind(IL)))) active#(take(s(M),cons(N,IL))) -> isNatKind#(N) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatKind(N)) active#(take(s(M),cons(N,IL))) -> isNatKind#(M) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNatKind(M)) active#(take(s(M),cons(N,IL))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(take(s(M),cons(N,IL))) -> isNatIListKind#(IL) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(s(M),cons(N,IL))) -> and#(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) active#(take(s(M),cons(N,IL))) -> U91#(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) active#(take(s(M),cons(N,IL))) -> mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> U12#(mark(X)) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> U21#(mark(X1),X2) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> U22#(mark(X)) mark#(U22(X)) -> active#(U22(mark(X))) mark#(isNat(X)) -> active#(isNat(X)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> U31#(mark(X1),X2) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> U32#(mark(X)) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> U41#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> U42#(mark(X1),X2) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> U43#(mark(X)) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> U52#(mark(X1),X2) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> U53#(mark(X)) mark#(U53(X)) -> active#(U53(mark(X))) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> U62#(mark(X1),X2) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> U63#(mark(X)) mark#(U63(X)) -> active#(U63(mark(X))) mark#(U71(X1,X2)) -> mark#(X1) mark#(U71(X1,X2)) -> U71#(mark(X1),X2) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> U81#(mark(X)) mark#(U81(X)) -> active#(U81(mark(X))) mark#(nil()) -> active#(nil()) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> U91#(mark(X1),X2,X3,X4) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) mark#(isNatKind(X)) -> active#(isNatKind(X)) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U12#(active(X)) -> U12#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U22#(active(X)) -> U22#(X) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U32#(active(X)) -> U32#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U42#(mark(X1),X2) -> U42#(X1,X2) U42#(X1,mark(X2)) -> U42#(X1,X2) U42#(active(X1),X2) -> U42#(X1,X2) U42#(X1,active(X2)) -> U42#(X1,X2) U43#(mark(X)) -> U43#(X) U43#(active(X)) -> U43#(X) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,mark(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) -> U51#(X1,X2,X3) U51#(active(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,active(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,active(X3)) -> U51#(X1,X2,X3) U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) U53#(mark(X)) -> U53#(X) U53#(active(X)) -> U53#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) U62#(active(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) U63#(mark(X)) -> U63#(X) U63#(active(X)) -> U63#(X) U71#(mark(X1),X2) -> U71#(X1,X2) U71#(X1,mark(X2)) -> U71#(X1,X2) U71#(active(X1),X2) -> U71#(X1,X2) U71#(X1,active(X2)) -> U71#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) U81#(mark(X)) -> U81#(X) U81#(active(X)) -> U81#(X) U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,mark(X2),X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,mark(X3),X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,X3,mark(X4)) -> U91#(X1,X2,X3,X4) U91#(active(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,active(X2),X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,active(X3),X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,X3,active(X4)) -> U91#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNatIListKind#(mark(X)) -> isNatIListKind#(X) isNatIListKind#(active(X)) -> isNatIListKind#(X) isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) TDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) active#(U11(tt(),V1)) -> isNatList#(V1) active#(U11(tt(),V1)) -> U12#(isNatList(V1)) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> isNatList#(V) active#(U31(tt(),V)) -> U32#(isNatList(V)) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) active#(U32(tt())) -> mark#(tt()) active#(U41(tt(),V1,V2)) -> isNat#(V1) active#(U41(tt(),V1,V2)) -> U42#(isNat(V1),V2) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) active#(U42(tt(),V2)) -> isNatIList#(V2) active#(U42(tt(),V2)) -> U43#(isNatIList(V2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) active#(U43(tt())) -> mark#(tt()) active#(U51(tt(),V1,V2)) -> isNat#(V1) active#(U51(tt(),V1,V2)) -> U52#(isNat(V1),V2) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> isNatList#(V2) active#(U52(tt(),V2)) -> U53#(isNatList(V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) active#(U53(tt())) -> mark#(tt()) active#(U61(tt(),V1,V2)) -> isNat#(V1) active#(U61(tt(),V1,V2)) -> U62#(isNat(V1),V2) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) active#(U62(tt(),V2)) -> isNatIList#(V2) active#(U62(tt(),V2)) -> U63#(isNatIList(V2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) active#(U63(tt())) -> mark#(tt()) active#(U71(tt(),L)) -> length#(L) active#(U71(tt(),L)) -> s#(length(L)) active#(U71(tt(),L)) -> mark#(s(length(L))) active#(U81(tt())) -> mark#(nil()) active#(U91(tt(),IL,M,N)) -> take#(M,IL) active#(U91(tt(),IL,M,N)) -> cons#(N,take(M,IL)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(length(V1))) -> isNatIListKind#(V1) active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) active#(isNatIList(V)) -> isNatIListKind#(V) active#(isNatIList(V)) -> U31#(isNatIListKind(V),V) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIList(cons(V1,V2))) -> U41#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(take(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatKind(0())) -> mark#(tt()) active#(isNatKind(length(V1))) -> isNatIListKind#(V1) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(cons(V1,V2))) -> U51#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(take(V1,V2))) -> isNatKind#(V1) active#(isNatList(take(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(take(V1,V2))) -> U61#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> isNatKind#(N) active#(length(cons(N,L))) -> isNat#(N) active#(length(cons(N,L))) -> and#(isNat(N),isNatKind(N)) active#(length(cons(N,L))) -> isNatIListKind#(L) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> and#(isNatList(L),isNatIListKind(L)) active#(length(cons(N,L))) -> and#(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) active#(length(cons(N,L))) -> U71#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active#(take(0(),IL)) -> isNatIListKind#(IL) active#(take(0(),IL)) -> isNatIList#(IL) active#(take(0(),IL)) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(0(),IL)) -> U81#(and(isNatIList(IL),isNatIListKind(IL))) active#(take(0(),IL)) -> mark#(U81(and(isNatIList(IL),isNatIListKind(IL)))) active#(take(s(M),cons(N,IL))) -> isNatKind#(N) active#(take(s(M),cons(N,IL))) -> isNat#(N) active#(take(s(M),cons(N,IL))) -> and#(isNat(N),isNatKind(N)) active#(take(s(M),cons(N,IL))) -> isNatKind#(M) active#(take(s(M),cons(N,IL))) -> isNat#(M) active#(take(s(M),cons(N,IL))) -> and#(isNat(M),isNatKind(M)) active#(take(s(M),cons(N,IL))) -> and#(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) active#(take(s(M),cons(N,IL))) -> isNatIListKind#(IL) active#(take(s(M),cons(N,IL))) -> isNatIList#(IL) active#(take(s(M),cons(N,IL))) -> and#(isNatIList(IL),isNatIListKind(IL)) active#(take(s(M),cons(N,IL))) -> and#(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))) active#(take(s(M),cons(N,IL))) -> U91#(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N) active#(take(s(M),cons(N,IL))) -> mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> U11#(mark(X1),X2) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) mark#(tt()) -> active#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> U12#(mark(X)) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> U21#(mark(X1),X2) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> U22#(mark(X)) mark#(U22(X)) -> active#(U22(mark(X))) mark#(isNat(X)) -> active#(isNat(X)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> U31#(mark(X1),X2) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> U32#(mark(X)) mark#(U32(X)) -> active#(U32(mark(X))) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> U41#(mark(X1),X2,X3) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> U42#(mark(X1),X2) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> U43#(mark(X)) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> U51#(mark(X1),X2,X3) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> U52#(mark(X1),X2) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> U53#(mark(X)) mark#(U53(X)) -> active#(U53(mark(X))) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> U62#(mark(X1),X2) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> U63#(mark(X)) mark#(U63(X)) -> active#(U63(mark(X))) mark#(U71(X1,X2)) -> mark#(X1) mark#(U71(X1,X2)) -> U71#(mark(X1),X2) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> U81#(mark(X)) mark#(U81(X)) -> active#(U81(mark(X))) mark#(nil()) -> active#(nil()) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> U91#(mark(X1),X2,X3,X4) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> and#(mark(X1),X2) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) mark#(isNatKind(X)) -> active#(isNatKind(X)) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U12#(active(X)) -> U12#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U22#(active(X)) -> U22#(X) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U32#(active(X)) -> U32#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) U42#(mark(X1),X2) -> U42#(X1,X2) U42#(X1,mark(X2)) -> U42#(X1,X2) U42#(active(X1),X2) -> U42#(X1,X2) U42#(X1,active(X2)) -> U42#(X1,X2) U43#(mark(X)) -> U43#(X) U43#(active(X)) -> U43#(X) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,mark(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) -> U51#(X1,X2,X3) U51#(active(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,active(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,active(X3)) -> U51#(X1,X2,X3) U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) U53#(mark(X)) -> U53#(X) U53#(active(X)) -> U53#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) U62#(active(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) U63#(mark(X)) -> U63#(X) U63#(active(X)) -> U63#(X) U71#(mark(X1),X2) -> U71#(X1,X2) U71#(X1,mark(X2)) -> U71#(X1,X2) U71#(active(X1),X2) -> U71#(X1,X2) U71#(X1,active(X2)) -> U71#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) U81#(mark(X)) -> U81#(X) U81#(active(X)) -> U81#(X) U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,mark(X2),X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,mark(X3),X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,X3,mark(X4)) -> U91#(X1,X2,X3,X4) U91#(active(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,active(X2),X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,active(X3),X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,X3,active(X4)) -> U91#(X1,X2,X3,X4) take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) isNatIListKind#(mark(X)) -> isNatIListKind#(X) isNatIListKind#(active(X)) -> isNatIListKind#(X) isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) graph: ... SCC Processor: #sccs: 29 #rules: 190 #arcs: 9356/81796 DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) mark#(tt()) -> active#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) active#(U81(tt())) -> mark#(nil()) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> mark#(X1) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(nil()) -> active#(nil()) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active#(take(0(),IL)) -> mark#(U81(and(isNatIList(IL),isNatIListKind(IL)))) active#(take(s(M),cons(N,IL))) -> mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [U91](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 1, [nil] = 0, [U81](x0) = x0, [s](x0) = x0, [length](x0) = x0, [U71](x0, x1) = x0 + x1, [U63](x0) = x0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U11(tt(),V1)) = 0 >= 0 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 0 >= 0 = mark#(tt()) mark#(0()) = 0 >= 0 = active#(0()) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= X1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U31(tt(),V)) = 0 >= 0 = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= X1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 0 >= 0 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= X1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 0 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= X1 = active#(U51(mark(X1),X2,X3)) active#(U81(tt())) = 0 >= 0 = mark#(nil()) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark#(cons(N,take(M,IL))) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= X1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= X1 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U71(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(isNatIList(zeros())) = 0 >= 0 = mark#(tt()) mark#(length(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = X >= X = mark#(X) mark#(U81(X)) = X >= X = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) mark#(nil()) = 0 >= 0 = active#(nil()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) mark#(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) 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 >= X1 + X2 + 1 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 0 >= 0 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(nil())) = 0 >= 0 = mark#(0()) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active#(take(0(),IL)) = IL + 1 >= 0 = mark#(U81(and(isNatIList(IL),isNatIListKind(IL)))) active#(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))), IL,M,N)) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X1 >= X1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 >= X1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 >= X1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 >= X1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 >= X1 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 >= X1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 >= X1 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = X1 + X2 >= X1 + X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = X >= X = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,mark(X2)) = X1 >= X1 = U11(X1,X2) U11(active(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,active(X2)) = X1 >= X1 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 >= X1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 >= X1 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,mark(X2)) = X1 >= X1 = U42(X1,X2) U42(active(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,active(X2)) = X1 >= X1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 >= X1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 >= X1 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,mark(X2)) = X1 >= X1 = U52(X1,X2) U52(active(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,active(X2)) = X1 >= X1 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 >= X1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 >= X1 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,mark(X2)) = X1 >= X1 = U62(X1,X2) U62(active(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,active(X2)) = X1 >= X1 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = X1 + X2 >= X1 + X2 = U71(X1,X2) U71(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U71(X1,X2) U71(active(X1),X2) = X1 + X2 >= X1 + X2 = U71(X1,X2) U71(X1,active(X2)) = X1 + X2 >= X1 + X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = X >= X = U81(X) U81(active(X)) = X >= X = U81(X) U91(mark(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X1 + X2 + X3 + X4 + 1 >= X1 + X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) mark#(tt()) -> active#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) active#(U81(tt())) -> mark#(nil()) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> mark#(X1) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(nil()) -> active#(nil()) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active#(take(s(M),cons(N,IL))) -> mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1, [U91](x0, x1, x2, x3) = x1 + x3, [nil] = 0, [U81](x0) = x0, [s](x0) = x0, [length](x0) = x0 + 1, [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, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U11(tt(),V1)) = 0 >= 0 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 0 >= 0 = mark#(tt()) mark#(0()) = 0 >= 0 = active#(0()) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= X1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U31(tt(),V)) = 0 >= 0 = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= X1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 0 >= 0 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= X1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 0 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = L + 1 >= L + 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= X1 = active#(U51(mark(X1),X2,X3)) active#(U81(tt())) = 0 >= 0 = mark#(nil()) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) = IL + N >= IL + N = mark#(cons(N,take(M,IL))) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= X1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= X1 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U71(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(isNatIList(zeros())) = 0 >= 0 = mark#(tt()) mark#(length(X)) = X + 1 >= X = mark#(X) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = X >= X = mark#(X) mark#(U81(X)) = X >= X = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) mark#(nil()) = 0 >= 0 = active#(nil()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) mark#(U91(X1,X2,X3,X4)) = X2 + X4 >= X2 + X4 = active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = X2 >= X2 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 0 >= 0 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(nil())) = 1 >= 0 = mark#(0()) active#(length(cons(N,L))) = L + N + 1 >= L + 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active#(take(s(M),cons(N,IL))) = IL + N >= IL + N = mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))), IL,M,N)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + N >= IL + N = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + N >= IL + N = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X1 >= X1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 >= X1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 >= X1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 >= X1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 >= X1 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 >= X1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 >= X1 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = active(length(mark(X))) mark(U81(X)) = X >= X = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X4 >= X2 + X4 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 >= X2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,mark(X2)) = X1 >= X1 = U11(X1,X2) U11(active(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,active(X2)) = X1 >= X1 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 >= X1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 >= X1 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,mark(X2)) = X1 >= X1 = U42(X1,X2) U42(active(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,active(X2)) = X1 >= X1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 >= X1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 >= X1 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,mark(X2)) = X1 >= X1 = U52(X1,X2) U52(active(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,active(X2)) = X1 >= X1 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 >= X1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 >= X1 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,mark(X2)) = X1 >= X1 = U62(X1,X2) U62(active(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,active(X2)) = X1 >= X1 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U71(X1,X2) U71(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U71(X1,X2) U71(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U71(X1,X2) U71(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = length(X) U81(mark(X)) = X >= X = U81(X) U81(active(X)) = X >= X = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X4 >= X2 + X4 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X4 >= X2 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X4 >= X2 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X4 >= X2 + X4 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X4 >= X2 + X4 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X4 >= X2 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X4 >= X2 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X4 >= X2 + X4 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 >= X2 = take(X1,X2) take(X1,mark(X2)) = X2 >= X2 = take(X1,X2) take(active(X1),X2) = X2 >= X2 = take(X1,X2) take(X1,active(X2)) = X2 >= X2 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) mark#(tt()) -> active#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) active#(U81(tt())) -> mark#(nil()) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(nil()) -> active#(nil()) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active#(take(s(M),cons(N,IL))) -> mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = x1 + x3 + 1, [nil] = 0, [U81](x0) = x0 + 1, [s](x0) = x0, [length](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, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U11(tt(),V1)) = 0 >= 0 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 0 >= 0 = mark#(tt()) mark#(0()) = 0 >= 0 = active#(0()) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= X1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(tt()) = 0 >= 0 = active#(tt()) active#(U31(tt(),V)) = 0 >= 0 = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= X1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 0 >= 0 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= X1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 0 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= X1 = active#(U51(mark(X1),X2,X3)) active#(U81(tt())) = 1 >= 0 = mark#(nil()) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) = IL + N + 1 >= IL + N + 1 = mark#(cons(N,take(M,IL))) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= X1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= X1 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(s(X)) = X >= X = active#(s(mark(X))) active#(isNatIList(zeros())) = 0 >= 0 = mark#(tt()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = X + 1 >= X = mark#(X) mark#(U81(X)) = X + 1 >= X + 1 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) mark#(nil()) = 0 >= 0 = active#(nil()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) mark#(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = X2 + 1 >= X2 + 1 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 0 >= 0 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active#(take(s(M),cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))), IL,M,N)) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + N + 1 >= IL + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X1 >= X1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 >= X1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 >= X1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 >= X1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 >= X1 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 >= X1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 >= X1 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = X + 1 >= X + 1 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,mark(X2)) = X1 >= X1 = U11(X1,X2) U11(active(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,active(X2)) = X1 >= X1 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 >= X1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 >= X1 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,mark(X2)) = X1 >= X1 = U42(X1,X2) U42(active(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,active(X2)) = X1 >= X1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 >= X1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 >= X1 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,mark(X2)) = X1 >= X1 = U52(X1,X2) U52(active(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,active(X2)) = X1 >= X1 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 >= X1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 >= X1 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,mark(X2)) = X1 >= X1 = U62(X1,X2) U62(active(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,active(X2)) = X1 >= X1 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = X + 1 >= X + 1 = U81(X) U81(active(X)) = X + 1 >= X + 1 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) take(active(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) mark#(tt()) -> active#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(nil()) -> active#(nil()) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active#(take(s(M),cons(N,IL))) -> mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [active#](x0) = x0 + 1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1, [U91](x0, x1, x2, x3) = x3 + 1, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](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, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + 1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 2 >= 2 = mark#(cons(0(),zeros())) mark#(zeros()) = 2 >= 2 = active#(zeros()) active#(U11(tt(),V1)) = 1 >= 1 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = mark#(X1) mark#(cons(X1,X2)) = X1 + 2 >= X1 + 2 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 1 >= 1 = mark#(tt()) mark#(0()) = 1 >= 1 = active#(0()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U11(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(tt()) = 1 >= 1 = active#(tt()) active#(U31(tt(),V)) = 1 >= 1 = mark#(U32(isNatList(V))) mark#(U12(X)) = X + 1 >= X + 1 = mark#(X) mark#(U12(X)) = X + 1 >= X + 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U21(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X + 1 >= X + 1 = mark#(X) mark#(U22(X)) = X + 1 >= X + 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U31(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X + 1 >= X + 1 = mark#(X) mark#(U32(X)) = X + 1 >= X + 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U42(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X + 1 >= X + 1 = mark#(X) mark#(U43(X)) = X + 1 >= X + 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U52(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) = N + 2 >= N + 2 = mark#(cons(N,take(M,IL))) mark#(U53(X)) = X + 1 >= X + 1 = mark#(X) mark#(U53(X)) = X + 1 >= X + 1 = active#(U53(mark(X))) active#(and(tt(),X)) = X + 1 >= X + 1 = mark#(X) mark#(U61(X1,X2,X3)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = X1 + 1 >= X1 + 1 = mark#(X1) mark#(U62(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = 1 >= 1 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = X + 1 >= X + 1 = mark#(X) mark#(U63(X)) = X + 1 >= X + 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) mark#(s(X)) = X + 1 >= X + 1 = active#(s(mark(X))) active#(isNatIList(zeros())) = 1 >= 1 = mark#(tt()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = 1 >= 1 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) mark#(nil()) = 1 >= 1 = active#(nil()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) mark#(U91(X1,X2,X3,X4)) = X4 + 2 >= X4 + 2 = active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = X2 + 1 >= X2 + 1 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active#(take(s(M),cons(N,IL))) = N + 2 >= N + 2 = mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))), IL,M,N)) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = N + 1 >= N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = N + 1 >= N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X1 >= X1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 >= X1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 >= X1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 >= X1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 >= X1 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 >= X1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 >= X1 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X4 + 1 >= X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 >= X2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(active(X1),X2) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(X1,active(X2)) = X1 + 1 >= X1 + 1 = cons(X1,X2) U11(mark(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,mark(X2)) = X1 >= X1 = U11(X1,X2) U11(active(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,active(X2)) = X1 >= X1 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 >= X1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 >= X1 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,mark(X2)) = X1 >= X1 = U42(X1,X2) U42(active(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,active(X2)) = X1 >= X1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 >= X1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 >= X1 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,mark(X2)) = X1 >= X1 = U52(X1,X2) U52(active(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,active(X2)) = X1 >= X1 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 >= X1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 >= X1 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,mark(X2)) = X1 >= X1 = U62(X1,X2) U62(active(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,active(X2)) = X1 >= X1 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X4 + 1 >= X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 >= X2 = take(X1,X2) take(X1,mark(X2)) = X2 >= X2 = take(X1,X2) take(active(X1),X2) = X2 >= X2 = take(X1,X2) take(X1,active(X2)) = X2 >= X2 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) mark#(tt()) -> active#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(nil()) -> active#(nil()) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active#(take(s(M),cons(N,IL))) -> mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 1, [U81](x0) = 1, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 1, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U11(tt(),V1)) = 1 >= 1 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 1 >= 1 = mark#(tt()) mark#(0()) = 1 >= 1 = active#(0()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2)) = 1 >= 1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(tt()) = 1 >= 0 = active#(tt()) active#(U31(tt(),V)) = 1 >= 1 = mark#(U32(isNatList(V))) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) = 1 >= 1 = mark#(cons(N,take(M,IL))) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = 1 >= 1 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(s(X)) = 1 >= 0 = active#(s(mark(X))) active#(isNatIList(zeros())) = 1 >= 1 = mark#(tt()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = 1 >= 1 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) mark#(nil()) = 1 >= 1 = active#(nil()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) mark#(U91(X1,X2,X3,X4)) = 1 >= 1 = active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = 1 >= 1 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active#(take(s(M),cons(N,IL))) = 1 >= 1 = mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))), IL,M,N)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(nil()) -> active#(nil()) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active#(take(s(M),cons(N,IL))) -> mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](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, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U11(tt(),V1)) = 0 >= 0 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = 0 >= 0 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 0 >= 0 = mark#(tt()) mark#(0()) = 0 >= 0 = active#(0()) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= X1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) active#(U31(tt(),V)) = 0 >= 0 = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= X1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 0 >= 0 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= X1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 0 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= X1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) = 0 >= 0 = mark#(cons(N,take(M,IL))) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= X1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= X1 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 0 >= 0 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) active#(isNatIList(zeros())) = 0 >= 0 = mark#(tt()) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = 0 >= 0 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) mark#(nil()) = 0 >= 0 = active#(nil()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) mark#(U91(X1,X2,X3,X4)) = 0 >= 0 = active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = 1 >= 1 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 0 >= 0 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 0 >= 0 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active#(take(s(M),cons(N,IL))) = 1 >= 0 = mark#(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and (isNat(N), isNatKind (N)))), IL,M,N)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X1 >= X1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 >= X1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 >= X1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 >= X1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 >= X1 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 >= X1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 >= X1 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,mark(X2)) = X1 >= X1 = U11(X1,X2) U11(active(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,active(X2)) = X1 >= X1 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 >= X1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 >= X1 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,mark(X2)) = X1 >= X1 = U42(X1,X2) U42(active(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,active(X2)) = X1 >= X1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 >= X1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 >= X1 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,mark(X2)) = X1 >= X1 = U52(X1,X2) U52(active(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,active(X2)) = X1 >= X1 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 >= X1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 >= X1 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,mark(X2)) = X1 >= X1 = U62(X1,X2) U62(active(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,active(X2)) = X1 >= X1 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL))) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(nil()) -> active#(nil()) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](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, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U11(tt(),V1)) = 0 >= 0 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = 0 >= 0 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 0 >= 0 = mark#(tt()) mark#(0()) = 0 >= 0 = active#(0()) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 >= X1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) active#(U31(tt(),V)) = 0 >= 0 = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 >= X1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 >= X1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 0 >= 0 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 >= X1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 0 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= X1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = active#(U52(mark(X1),X2)) active#(U91(tt(),IL,M,N)) = 1 >= 0 = mark#(cons(N,take(M,IL))) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 >= X1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 >= X1 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 0 >= 0 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) active#(isNatIList(zeros())) = 0 >= 0 = mark#(tt()) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = 0 >= 0 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) mark#(nil()) = 0 >= 0 = active#(nil()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) mark#(U91(X1,X2,X3,X4)) = 1 >= 1 = active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = 1 >= 1 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 0 >= 0 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 0 >= 0 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X1 >= X1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = X1 >= X1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 >= X1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 >= X1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 >= X1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 >= X1 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 >= X1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 >= X1 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,mark(X2)) = X1 >= X1 = U11(X1,X2) U11(active(X1),X2) = X1 >= X1 = U11(X1,X2) U11(X1,active(X2)) = X1 >= X1 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,mark(X2)) = X1 >= X1 = U31(X1,X2) U31(active(X1),X2) = X1 >= X1 = U31(X1,X2) U31(X1,active(X2)) = X1 >= X1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 >= X1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 >= X1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 >= X1 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,mark(X2)) = X1 >= X1 = U42(X1,X2) U42(active(X1),X2) = X1 >= X1 = U42(X1,X2) U42(X1,active(X2)) = X1 >= X1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 >= X1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 >= X1 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,mark(X2)) = X1 >= X1 = U52(X1,X2) U52(active(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,active(X2)) = X1 >= X1 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 >= X1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 >= X1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 >= X1 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,mark(X2)) = X1 >= X1 = U62(X1,X2) U62(active(X1),X2) = X1 >= X1 = U62(X1,X2) U62(X1,active(X2)) = X1 >= X1 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(nil()) -> active#(nil()) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and( and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 1, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U11(tt(),V1)) = 1 >= 1 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 1 >= 1 = mark#(tt()) mark#(0()) = 1 >= 1 = active#(0()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2)) = 1 >= 1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) active#(U31(tt(),V)) = 1 >= 1 = mark#(U32(isNatList(V))) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = 1 >= 1 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) active#(isNatIList(zeros())) = 1 >= 1 = mark#(tt()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = 1 >= 1 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) mark#(nil()) = 1 >= 0 = active#(nil()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) mark#(U91(X1,X2,X3,X4)) = 1 >= 0 = active#(U91(mark(X1),X2,X3,X4)) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = 1 >= 1 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and (and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U91](x0, x1, x2, x3) = x1 + x2 + x3, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = x0 + 1, [U71](x0, x1) = x1 + 1, [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, [U43](x0) = x0, [isNatIList](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, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U11(tt(),V1)) = V1 >= V1 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 0 >= 0 = mark#(tt()) mark#(0()) = 0 >= 0 = active#(0()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) active#(U31(tt(),V)) = V >= V = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = V2 >= V2 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = V2 >= V2 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = X >= X = active#(isNatIList(X)) active#(U71(tt(),L)) = L + 1 >= L + 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U62(mark(X1),X2)) active#(isNat(length(V1))) = V1 + 1 >= V1 = mark#(U11(isNatIListKind(V1),V1)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = V >= V = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) active#(isNatIList(zeros())) = 0 >= 0 = mark#(tt()) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = 0 >= 0 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = X1 + X2 >= X1 + X2 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N + 1 >= L + 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= V1 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= V = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= V2 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = V2 >= V2 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + M + N >= IL + M + N = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 1 >= V1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= V = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + M + N >= IL + M + N = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X1 + X2 >= X1 + X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 + X2 >= X1 + X2 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = X >= X = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 + X2 >= X1 + X2 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = X2 + 1 >= X2 + 1 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(active(X1),X2) = X1 + X2 >= X1 + X2 = U11(X1,X2) U11(X1,active(X2)) = X1 + X2 >= X1 + X2 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(active(X1),X2) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(X1,active(X2)) = X1 + X2 >= X1 + X2 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = X >= X = isNatIList(X) isNatIList(active(X)) = X >= X = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(active(X1),X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(X1,active(X2)) = X1 + X2 >= X1 + X2 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(active(X1),X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(X1,active(X2)) = X2 + 1 >= X2 + 1 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and (and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 1, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U11(tt(),V1)) = 1 >= 1 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 1 >= 1 = mark#(tt()) mark#(0()) = 1 >= 1 = active#(0()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2)) = 1 >= 1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) active#(U31(tt(),V)) = 1 >= 1 = mark#(U32(isNatList(V))) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) active#(isNatIList(zeros())) = 1 >= 1 = mark#(tt()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = 1 >= 1 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(take(X1,X2)) = 1 >= 0 = active#(take(mark(X1),mark(X2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) -> active#(U81(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and ( and(isNat(M),isNatKind(M)), and (isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 1, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U11(tt(),V1)) = 1 >= 1 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 1 >= 1 = mark#(tt()) mark#(0()) = 1 >= 1 = active#(0()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2)) = 1 >= 1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) active#(U31(tt(),V)) = 1 >= 1 = mark#(U32(isNatList(V))) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) active#(isNatIList(zeros())) = 1 >= 1 = mark#(tt()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(U81(X)) = 1 >= 0 = active#(U81(mark(X))) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) mark#(0()) -> active#(0()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U11(tt(),V1)) = 1 >= 1 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 1 >= 1 = mark#(tt()) mark#(0()) = 1 >= 0 = active#(0()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U11(X1,X2)) = 1 >= 1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) active#(U31(tt(),V)) = 1 >= 1 = mark#(U32(isNatList(V))) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) active#(isNatIList(zeros())) = 1 >= 1 = mark#(tt()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U11(tt(),V1)) -> mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> mark#(X1) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U91](x0, x1, x2, x3) = x1 + x2 + x3, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = x0 + 1, [U71](x0, x1) = x1 + 1, [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, [U43](x0) = x0, [isNatIList](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, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0 + x1 + 1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U11(tt(),V1)) = V1 + 1 >= V1 = mark#(U12(isNatList(V1))) mark#(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 0 >= 0 = mark#(tt()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 0 >= 0 = mark#(tt()) active#(U31(tt(),V)) = V >= V = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = V2 >= V2 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = V2 >= V2 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = X >= X = active#(isNatIList(X)) active#(U71(tt(),L)) = L + 1 >= L + 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U62(mark(X1),X2)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = V >= V = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) active#(isNatIList(zeros())) = 0 >= 0 = mark#(tt()) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N + 1 >= L + 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 + 1 >= V1 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= V = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= V2 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = V2 >= V2 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + M + N >= IL + M + N = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 1 >= V1 + 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= V = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + M + N >= IL + M + N = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 + X2 >= X1 + X2 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = X >= X = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 + X2 >= X1 + X2 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = X2 + 1 >= X2 + 1 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(X1,X2) U11(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(X1,X2) U11(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(X1,X2) U11(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(active(X1),X2) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(X1,active(X2)) = X1 + X2 >= X1 + X2 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = X >= X = isNatIList(X) isNatIList(active(X)) = X >= X = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(active(X1),X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(X1,active(X2)) = X1 + X2 >= X1 + X2 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(active(X1),X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(X1,active(X2)) = X2 + 1 >= X2 + 1 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2)) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 1 >= 1 = mark#(tt()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) mark#(U11(X1,X2)) = 1 >= 0 = active#(U11(mark(X1),X2)) active#(U22(tt())) = 1 >= 1 = mark#(tt()) active#(U31(tt(),V)) = 1 >= 1 = mark#(U32(isNatList(V))) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) active#(isNatIList(zeros())) = 1 >= 1 = mark#(tt()) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) active#(isNatIList(zeros())) -> mark#(tt()) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U91](x0, x1, x2, x3) = x1 + x2 + x3, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = x0, [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, [U43](x0) = x0, [isNatIList](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, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) mark#(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 0 >= 0 = mark#(tt()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) active#(U31(tt(),V)) = V >= V = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = V2 >= V2 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = V2 >= V2 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = X >= X = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U62(mark(X1),X2)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 >= X2 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = V >= V = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) active#(isNatIList(zeros())) = 1 >= 0 = mark#(tt()) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= V1 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= V = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= V2 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = V2 >= V2 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + M + N >= IL + M + N = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= V = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + M + N >= IL + M + N = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X2 >= X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = X1 + X2 >= X1 + X2 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 + X2 >= X1 + X2 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = X >= X = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 + X2 >= X1 + X2 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = X2 >= X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,mark(X2)) = X2 >= X2 = U11(X1,X2) U11(active(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,active(X2)) = X2 >= X2 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 >= X1 + X2 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 >= X1 + X2 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(active(X1),X2) = X1 + X2 >= X1 + X2 = U42(X1,X2) U42(X1,active(X2)) = X1 + X2 >= X1 + X2 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = X >= X = isNatIList(X) isNatIList(active(X)) = X >= X = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(active(X1),X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(X1,active(X2)) = X1 + X2 >= X1 + X2 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,mark(X2)) = X2 >= X2 = U71(X1,X2) U71(active(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,active(X2)) = X2 >= X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 1, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) mark#(cons(X1,X2)) = 1 >= 0 = active#(cons(mark(X1),X2)) active#(U12(tt())) = 1 >= 1 = mark#(tt()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 1 >= 1 = mark#(tt()) active#(U31(tt(),V)) = 1 >= 1 = mark#(U32(isNatList(V))) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X1,X2)) = 1 >= 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) active#(U31(tt(),V)) -> mark#(U32(isNatList(V))) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> mark#(X1) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = x0, [U71](x0, x1) = x1, [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, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = x0 + 1, [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, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U12(tt())) = 0 >= 0 = mark#(tt()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) active#(U31(tt(),V)) = V + 1 >= V = mark#(U32(isNatList(V))) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = V2 + 1 >= V2 + 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U42(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = V2 + 1 >= V2 + 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = X + 1 >= X + 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),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 >= X1 + X2 + X3 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 >= X2 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = V + 1 >= V + 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= V1 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V + 1 >= V = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 + 1 >= V2 + 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = V2 + 1 >= V2 + 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= V + 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V1 + V2 + 1 >= V1 + V2 + 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X2 >= X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = X2 >= X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,mark(X2)) = X2 >= X2 = U11(X1,X2) U11(active(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,active(X2)) = X2 >= X2 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U31(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(X1,X2,X3) U42(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(X1,X2) U42(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(X1,X2) U42(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(X1,X2) U42(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = X + 1 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 1 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) U62(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) U62(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) U62(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,mark(X2)) = X2 >= X2 = U71(X1,X2) U71(active(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,active(X2)) = X2 >= X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 0, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U12(tt())) = 1 >= 1 = mark#(tt()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) mark#(U31(X1,X2)) = 1 >= 0 = active#(U31(mark(X1),X2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = 1 >= 1 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) -> mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = x1 + x3 + 1, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = x0, [U71](x0, x1) = x1, [U63](x0) = x0, [U62](x0, x1) = x1, [U61](x0, x1, x2) = x2 + 1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x2, [U32](x0) = x0, [U31](x0, x1) = x1, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U12(tt())) = 0 >= 0 = mark#(tt()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = V2 >= V2 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = V2 >= V2 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X >= X = mark#(X) mark#(U32(X)) = X >= X = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X3 >= X3 = active#(U41(mark(X1),X2,X3)) active#(U61(tt(),V1,V2)) = V2 + 1 >= V2 = mark#(U62(isNat(V1),V2)) mark#(U42(X1,X2)) = X2 >= X2 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = V2 >= V2 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = X >= X = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) mark#(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X2 >= X2 = active#(U62(mark(X1),X2)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 >= X2 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = V >= V = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = V1 + V2 >= V2 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = V2 + 1 >= V2 + 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= V1 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= V = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V2 >= V2 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= V2 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = V2 + 1 >= V2 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = V2 >= V2 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + N + 1 >= IL + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= V = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 >= V2 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V2 + 1 >= V2 + 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X2 >= X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = X2 >= X2 = active(U31(mark(X1),X2)) mark(U32(X)) = X >= X = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X3 >= X3 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X2 >= X2 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = X >= X = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X2 >= X2 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = X2 >= X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,mark(X2)) = X2 >= X2 = U11(X1,X2) U11(active(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,active(X2)) = X2 >= X2 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,mark(X2)) = X2 >= X2 = U31(X1,X2) U31(active(X1),X2) = X2 >= X2 = U31(X1,X2) U31(X1,active(X2)) = X2 >= X2 = U31(X1,X2) U32(mark(X)) = X >= X = U32(X) U32(active(X)) = X >= X = U32(X) U41(mark(X1),X2,X3) = X3 >= X3 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X3 >= X3 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X3 >= X3 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X3 >= X3 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X3 >= X3 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X3 >= X3 = U41(X1,X2,X3) U42(mark(X1),X2) = X2 >= X2 = U42(X1,X2) U42(X1,mark(X2)) = X2 >= X2 = U42(X1,X2) U42(active(X1),X2) = X2 >= X2 = U42(X1,X2) U42(X1,active(X2)) = X2 >= X2 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = X >= X = isNatIList(X) isNatIList(active(X)) = X >= X = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2) = X2 >= X2 = U62(X1,X2) U62(X1,mark(X2)) = X2 >= X2 = U62(X1,X2) U62(active(X1),X2) = X2 >= X2 = U62(X1,X2) U62(X1,active(X2)) = X2 >= X2 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,mark(X2)) = X2 >= X2 = U71(X1,X2) U71(active(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,active(X2)) = X2 >= X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) take(active(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 0, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 0, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 1, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U12(tt())) = 1 >= 1 = mark#(tt()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X)) = 1 >= 1 = mark#(X) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = mark#(X) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 1 = mark#(tt()) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) mark#(U61(X1,X2,X3)) = 1 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 1 >= 1 = mark#(X) mark#(U63(X)) = 1 >= 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 1 = mark#(tt()) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = 1 >= 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 1 >= 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 1 >= 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) active#(U32(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> mark#(X) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = x1 + x3 + 1, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = x0, [U71](x0, x1) = x1, [U63](x0) = x0, [U62](x0, x1) = x1 + 1, [U61](x0, x1, x2) = x2 + 1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = x1 + 1, [U41](x0, x1, x2) = x2 + 1, [U32](x0) = x0 + 1, [U31](x0, x1) = x1 + 1, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U12(tt())) = 0 >= 0 = mark#(tt()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) active#(U32(tt())) = 1 >= 0 = mark#(tt()) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = V2 + 1 >= V2 + 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = X + 1 >= X = mark#(X) mark#(U32(X)) = X + 1 >= X + 1 = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = V2 + 1 >= V2 + 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatIList(X)) = X + 1 >= X + 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = X >= X = mark#(X) mark#(U63(X)) = X >= X = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 >= X2 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = V + 1 >= V + 1 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = V1 + V2 + 1 >= V2 + 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = V2 + 1 >= V2 + 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= V1 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V + 1 >= V + 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 + 1 >= V2 + 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = V2 + 1 >= V2 + 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + N + 1 >= IL + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= V + 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 + 1 >= V2 + 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V2 + 1 >= V2 + 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X2 >= X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = X2 + 1 >= X2 + 1 = active(U31(mark(X1),X2)) mark(U32(X)) = X + 1 >= X + 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X2 + 1 >= X2 + 1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X2 + 1 >= X2 + 1 = active(U62(mark(X1),X2)) mark(U63(X)) = X >= X = active(U63(mark(X))) mark(U71(X1,X2)) = X2 >= X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,mark(X2)) = X2 >= X2 = U11(X1,X2) U11(active(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,active(X2)) = X2 >= X2 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(active(X1),X2) = X2 + 1 >= X2 + 1 = U31(X1,X2) U31(X1,active(X2)) = X2 + 1 >= X2 + 1 = U31(X1,X2) U32(mark(X)) = X + 1 >= X + 1 = U32(X) U32(active(X)) = X + 1 >= X + 1 = U32(X) U41(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U41(X1,X2,X3) U42(mark(X1),X2) = X2 + 1 >= X2 + 1 = U42(X1,X2) U42(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U42(X1,X2) U42(active(X1),X2) = X2 + 1 >= X2 + 1 = U42(X1,X2) U42(X1,active(X2)) = X2 + 1 >= X2 + 1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = X + 1 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 1 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2) = X2 + 1 >= X2 + 1 = U62(X1,X2) U62(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U62(X1,X2) U62(active(X1),X2) = X2 + 1 >= X2 + 1 = U62(X1,X2) U62(X1,active(X2)) = X2 + 1 >= X2 + 1 = U62(X1,X2) U63(mark(X)) = X >= X = U63(X) U63(active(X)) = X >= X = U63(X) U71(mark(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,mark(X2)) = X2 >= X2 = U71(X1,X2) U71(active(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,active(X2)) = X2 >= X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) take(active(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(U63(tt())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> mark#(X) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = x1 + x3 + 1, [nil] = 0, [U81](x0) = 1, [s](x0) = x0, [length](x0) = x0, [U71](x0, x1) = x1, [U63](x0) = x0 + 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = x2 + 1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U12(tt())) = 0 >= 0 = mark#(tt()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 0 >= 0 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) active#(U63(tt())) = 1 >= 0 = mark#(tt()) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = X + 1 >= X = mark#(X) mark#(U63(X)) = X + 1 >= X + 1 = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 >= X2 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = V2 + 1 >= V2 + 1 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= V1 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = V2 + 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 1 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + N + 1 >= IL + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V2 + 1 >= V2 + 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X2 >= X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = X + 1 >= X + 1 = active(U63(mark(X))) mark(U71(X1,X2)) = X2 >= X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,mark(X2)) = X2 >= X2 = U11(X1,X2) U11(active(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,active(X2)) = X2 >= X2 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = X + 1 >= X + 1 = U63(X) U63(active(X)) = X + 1 >= X + 1 = U63(X) U71(mark(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,mark(X2)) = X2 >= X2 = U71(X1,X2) U71(active(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,active(X2)) = X2 >= X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) take(active(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) -> mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = x1 + x3 + 1, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = x0, [U71](x0, x1) = x1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U12(tt())) = 0 >= 0 = mark#(tt()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = X >= X = mark#(X) mark#(U12(X)) = X >= X = active#(U12(mark(X))) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 0 >= 0 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 0 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = 0 >= 0 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 0 >= 0 = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 >= X2 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 0 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatList(take(V1,V2))) = V2 + 1 >= 0 = mark#(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= V1 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + N + 1 >= IL + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= V1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V2 + 1 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X2 >= X2 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X >= X = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = X2 >= X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,mark(X2)) = X2 >= X2 = U11(X1,X2) U11(active(X1),X2) = X2 >= X2 = U11(X1,X2) U11(X1,active(X2)) = X2 >= X2 = U11(X1,X2) U12(mark(X)) = X >= X = U12(X) U12(active(X)) = X >= X = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,mark(X2)) = X2 >= X2 = U71(X1,X2) U71(active(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,active(X2)) = X2 >= X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) take(active(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U12(tt())) -> mark#(tt()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = x1 + x3 + 1, [nil] = 1, [U81](x0) = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U71](x0, x1) = x1 + 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0 + 1, [isNatList](x0) = x0, [U11](x0, x1) = x1 + 1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U12(tt())) = 1 >= 0 = mark#(tt()) active#(U21(tt(),V1)) = V1 >= V1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = X + 1 >= X = mark#(X) mark#(U12(X)) = X + 1 >= X + 1 = active#(U12(mark(X))) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 0 >= 0 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 0 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = L + 1 >= L + 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = 0 >= 0 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 0 >= 0 = active#(U63(mark(X))) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(nil())) = 1 >= 0 = mark#(tt()) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N + 1 >= L + 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 + 1 >= V1 + 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= V1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= V1 + V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + N + 1 >= IL + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 1 >= V1 + 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= V1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V2 + 1 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = X2 + 1 >= X2 + 1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = X + 1 >= X + 1 = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = X >= X = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = X2 + 1 >= X2 + 1 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = X2 + 1 >= X2 + 1 = U11(X1,X2) U11(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U11(X1,X2) U11(active(X1),X2) = X2 + 1 >= X2 + 1 = U11(X1,X2) U11(X1,active(X2)) = X2 + 1 >= X2 + 1 = U11(X1,X2) U12(mark(X)) = X + 1 >= X + 1 = U12(X) U12(active(X)) = X + 1 >= X + 1 = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(active(X1),X2) = X1 + X2 >= X1 + X2 = U21(X1,X2) U21(X1,active(X2)) = X1 + X2 >= X1 + X2 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(active(X1),X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(X1,active(X2)) = X2 + 1 >= X2 + 1 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) take(active(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = 1, [nil] = 1, [U81](x0) = 1, [s](x0) = x0, [length](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = 0 >= 0 = active#(U12(mark(X))) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 0 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 >= X1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 >= X1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = 0 >= 0 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 0 >= 0 = active#(U63(mark(X))) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 0 >= 0 = active#(U71(mark(X1),X2)) active#(isNatIList(V)) = 1 >= 0 = mark#(U31(isNatIListKind(V),V)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 0 >= 0 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 >= X1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 >= X1 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 >= X1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 >= X1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 >= X1 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,mark(X2)) = X1 >= X1 = U52(X1,X2) U52(active(X1),X2) = X1 >= X1 = U52(X1,X2) U52(X1,active(X2)) = X1 >= X1 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) take(active(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(U63(X)) -> active#(U63(mark(X))) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 0, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 0, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 1 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) mark#(U63(X)) = 1 >= 0 = active#(U63(mark(X))) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) -> mark#(U63(isNatIList(V2))) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x1 + 1, [U91](x0, x1, x2, x3) = x1 + x3 + 1, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x2, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = 0 >= 0 = active#(U12(mark(X))) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = V2 >= V2 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = V2 >= V2 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 0 >= 0 = active#(U42(mark(X1),X2)) active#(U62(tt(),V2)) = 1 >= 0 = mark#(U63(isNatIList(V2))) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + X3 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + X3 >= X1 + X3 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(0())) = 0 >= 0 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 0 >= 0 = active#(U71(mark(X1),X2)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V2 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 0 >= 0 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V2 >= V2 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= V2 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + N + 1 >= IL + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = V2 + 1 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + N + 1 >= IL + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + X3 >= X1 + X3 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + X2 >= X1 + X2 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X2 + 1 >= X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 + X3 >= X1 + X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + X3 >= X1 + X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + X3 >= X1 + X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + X3 >= X1 + X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + X3 >= X1 + X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + X3 >= X1 + X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(active(X1),X2) = X1 + X2 >= X1 + X2 = U52(X1,X2) U52(X1,active(X2)) = X1 + X2 >= X1 + X2 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X4 + 1 >= X2 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) take(active(X1),X2) = X2 + 1 >= X2 + 1 = take(X1,X2) take(X1,active(X2)) = X2 + 1 >= X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 1, [U31](x0, x1) = 0, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 1 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNat(0())) = 1 >= 1 = mark#(tt()) mark#(U62(X1,X2)) = 1 >= 0 = active#(U62(mark(X1),X2)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 1 >= 1 = U32(X) U32(active(X)) = 1 >= 1 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(zeros()) -> mark#(cons(0(),zeros())) mark#(zeros()) -> active#(zeros()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = 0, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = x0, [U52](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) active#(zeros()) = 1 >= 0 = mark#(cons(0(),zeros())) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = 0 >= 0 = active#(U12(mark(X))) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 >= X1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 0 >= 0 = active#(U42(mark(X1),X2)) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = X1 + 1 >= X1 = mark#(X1) mark#(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U52(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(0())) = 0 >= 0 = mark#(tt()) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 0 >= 0 = active#(U71(mark(X1),X2)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 0 >= 0 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 1 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U21(X1,X2)) = X1 >= X1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X1 + 1 >= X1 + 1 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,mark(X2)) = X1 >= X1 = U21(X1,X2) U21(active(X1),X2) = X1 >= X1 = U21(X1,X2) U21(X1,active(X2)) = X1 >= X1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X1 + 1 >= X1 + 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X1 + 1 >= X1 + 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X1 + 1 >= X1 + 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X1 + 1 >= X1 + 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X1 + 1 >= X1 + 1 = U51(X1,X2,X3) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = U52(X1,X2) U52(X1,mark(X2)) = X1 + 1 >= X1 + 1 = U52(X1,X2) U52(active(X1),X2) = X1 + 1 >= X1 + 1 = U52(X1,X2) U52(X1,active(X2)) = X1 + 1 >= X1 + 1 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(zeros()) -> active#(zeros()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> mark#(X1) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(0())) -> mark#(tt()) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](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, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = 1, [U21](x0, x1) = x0 + 1, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 0 >= 0 = active#(isNatKind(X)) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = 0 >= 0 = active#(U12(mark(X))) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U21(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 0 >= 0 = active#(U32(mark(X))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 0 >= 0 = active#(U42(mark(X1),X2)) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 0 >= 0 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 0 >= 0 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(0())) = 1 >= 0 = mark#(tt()) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 0 >= 0 = active#(U71(mark(X1),X2)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = 0 >= 0 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 0 >= 0 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 0 >= 0 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 0 >= 0 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = 1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = M >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = X1 + 1 >= X1 + 1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 >= X1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = U21(X1,X2) U21(X1,mark(X2)) = X1 + 1 >= X1 + 1 = U21(X1,X2) U21(active(X1),X2) = X1 + 1 >= X1 + 1 = U21(X1,X2) U21(X1,active(X2)) = X1 + 1 >= X1 + 1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 >= X1 = take(X1,X2) take(X1,mark(X2)) = X1 >= X1 = take(X1,X2) take(active(X1),X2) = X1 >= X1 = take(X1,X2) take(X1,active(X2)) = X1 >= X1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) mark#(zeros()) -> active#(zeros()) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) mark#(U32(X)) -> active#(U32(mark(X))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 1, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) mark#(zeros()) = 1 >= 0 = active#(zeros()) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X)) = 1 >= 1 = active#(U12(mark(X))) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) mark#(U32(X)) = 1 >= 0 = active#(U32(mark(X))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 1 >= 1 = U63(X) U63(active(X)) = 1 >= 1 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) -> mark#(tt()) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1 + 1, [U91](x0, x1, x2, x3) = x1 + x2 + x3 + 1, [nil] = 1, [U81](x0) = 1, [s](x0) = x0, [length](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, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x2, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(U12(X)) = 0 >= 0 = active#(U12(mark(X))) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = V2 >= V2 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 0 >= 0 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = V2 >= V2 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = X3 >= X3 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = X2 >= X2 = active#(U42(mark(X1),X2)) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = X >= X = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 0 >= 0 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 0 >= 0 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 >= X2 = active#(U71(mark(X1),X2)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = V1 + V2 >= V2 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(nil())) = 1 >= 0 = mark#(tt()) active#(isNatIListKind(zeros())) = 0 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(and(isNatKind(V1),isNatIListKind(V2))) active#(isNatIListKind(take(V1,V2))) = V1 + V2 + 1 >= V1 + V2 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = X >= X = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = V1 >= V1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = V1 >= V1 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V2 >= V2 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= V2 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + M + N + 1 >= IL + M + N + 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 >= V2 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = V1 + V2 + 1 >= V1 + V2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= V1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= V1 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL + 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + M + N + 1 >= IL + M + N + 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = X3 >= X3 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = X2 >= X2 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = X >= X = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = X2 >= X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = X >= X = active(isNatIListKind(X)) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = X3 >= X3 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = X3 >= X3 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = X3 >= X3 = U41(X1,X2,X3) U41(active(X1),X2,X3) = X3 >= X3 = U41(X1,X2,X3) U41(X1,active(X2),X3) = X3 >= X3 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = X3 >= X3 = U41(X1,X2,X3) U42(mark(X1),X2) = X2 >= X2 = U42(X1,X2) U42(X1,mark(X2)) = X2 >= X2 = U42(X1,X2) U42(active(X1),X2) = X2 >= X2 = U42(X1,X2) U42(X1,active(X2)) = X2 >= X2 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = X >= X = isNatIList(X) isNatIList(active(X)) = X >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,mark(X2)) = X2 >= X2 = U71(X1,X2) U71(active(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,active(X2)) = X2 >= X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X3 + X4 + 1 >= X2 + X3 + X4 + 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = X >= X = isNatIListKind(X) isNatIListKind(active(X)) = X >= X = isNatIListKind(X) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(U12(X)) -> active#(U12(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = 1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 1, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 1, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 1, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: mark#(isNatKind(X)) = 1 >= 1 = active#(isNatKind(X)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 1 >= 1 = mark#(tt()) mark#(U12(X)) = 1 >= 0 = active#(U12(mark(X))) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 1 >= 1 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 1 >= 1 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = 1 >= 1 = mark#(X) mark#(U22(X)) = 1 >= 1 = active#(U22(mark(X))) active#(U43(tt())) = 1 >= 1 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 1 >= 1 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 1 >= 1 = mark#(U53(isNatList(V2))) active#(U53(tt())) = 1 >= 1 = mark#(tt()) mark#(U41(X1,X2,X3)) = 1 >= 1 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 1 >= 1 = active#(U42(mark(X1),X2)) mark#(U43(X)) = 1 >= 1 = mark#(X) mark#(U43(X)) = 1 >= 1 = active#(U43(mark(X))) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U71(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 1 >= 1 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 1 >= 1 = active#(U52(mark(X1),X2)) mark#(U53(X)) = 1 >= 1 = mark#(X) mark#(U53(X)) = 1 >= 1 = active#(U53(mark(X))) active#(and(tt(),X)) = 1 >= 1 = mark#(X) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = 1 >= 1 = active#(U71(mark(X1),X2)) mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(zeros())) = 1 >= 1 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) mark#(and(X1,X2)) = 1 >= 1 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 1 >= 1 = mark#(tt()) mark#(isNatIListKind(X)) = 1 >= 1 = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = 1 >= 1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = 1 >= 1 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = 1 >= 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = 0 >= 0 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 0 >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 0 >= 0 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = 0 >= 0 = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = 0 >= 0 = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = 0 >= 0 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = 0 >= 0 = active(U71(mark(X1),X2)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = 0 >= 0 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 0 >= 0 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = 0 >= 0 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 0 >= 0 = active(isNatIListKind(X)) mark(isNatKind(X)) = 0 >= 0 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 1 >= 1 = U22(X) U22(active(X)) = 1 >= 1 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 1 >= 1 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 1 >= 1 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 1 >= 1 = U41(X1,X2,X3) U42(mark(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,mark(X2)) = 1 >= 1 = U42(X1,X2) U42(active(X1),X2) = 1 >= 1 = U42(X1,X2) U42(X1,active(X2)) = 1 >= 1 = U42(X1,X2) U43(mark(X)) = 1 >= 1 = U43(X) U43(active(X)) = 1 >= 1 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,mark(X2)) = 1 >= 1 = U52(X1,X2) U52(active(X1),X2) = 1 >= 1 = U52(X1,X2) U52(X1,active(X2)) = 1 >= 1 = U52(X1,X2) U53(mark(X)) = 1 >= 1 = U53(X) U53(active(X)) = 1 >= 1 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,mark(X2)) = 1 >= 1 = and(X1,X2) and(active(X1),X2) = 1 >= 1 = and(X1,X2) and(X1,active(X2)) = 1 >= 1 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(zeros())) -> mark#(tt()) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U91](x0, x1, x2, x3) = x1 + x2 + x3, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](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, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) active#(U21(tt(),V1)) = 1 >= 1 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 1 >= 1 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 0 >= 0 = active#(U42(mark(X1),X2)) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 0 >= 0 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 0 >= 0 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 >= X2 = active#(U71(mark(X1),X2)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(zeros())) = 1 >= 0 = mark#(tt()) active#(isNatIListKind(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = X >= X = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = V1 >= V1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = V1 >= V1 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N >= L = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L >= L = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + M + N >= IL + M + N = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = V1 + V2 >= V1 + V2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= V1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= V1 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= L = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + M + N >= IL + M + N = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = X2 >= X2 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = X >= X = active(isNatIListKind(X)) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,mark(X2)) = X2 >= X2 = U71(X1,X2) U71(active(X1),X2) = X2 >= X2 = U71(X1,X2) U71(X1,active(X2)) = X2 >= X2 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = X >= X = isNatIListKind(X) isNatIListKind(active(X)) = X >= X = isNatIListKind(X) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(length(V1))) -> mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0 + x1, [take](x0, x1) = x0 + x1, [U91](x0, x1, x2, x3) = x1 + x2 + x3, [nil] = 0, [U81](x0) = 0, [s](x0) = x0, [length](x0) = 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) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(isNatKind(X)) = X >= X = active#(isNatKind(X)) active#(U21(tt(),V1)) = 0 >= 0 = mark#(U22(isNat(V1))) active#(U22(tt())) = 0 >= 0 = mark#(tt()) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(U41(tt(),V1,V2)) = 0 >= 0 = mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) = 0 >= 0 = active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) = 0 >= 0 = mark#(U43(isNatIList(V2))) mark#(U22(X)) = X >= X = mark#(X) mark#(U22(X)) = X >= X = active#(U22(mark(X))) active#(U43(tt())) = 0 >= 0 = mark#(tt()) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(U51(tt(),V1,V2)) = 0 >= 0 = mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) = 0 >= 0 = mark#(U53(isNatList(V2))) active#(U53(tt())) = 0 >= 0 = mark#(tt()) mark#(U41(X1,X2,X3)) = 0 >= 0 = active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) = 0 >= 0 = active#(U42(mark(X1),X2)) mark#(U43(X)) = X >= X = mark#(X) mark#(U43(X)) = X >= X = active#(U43(mark(X))) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U71(tt(),L)) = L + 1 >= L + 1 = mark#(s(length(L))) mark#(U51(X1,X2,X3)) = 0 >= 0 = active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) = 0 >= 0 = active#(U52(mark(X1),X2)) mark#(U53(X)) = X >= X = mark#(X) mark#(U53(X)) = X >= X = active#(U53(mark(X))) active#(and(tt(),X)) = X >= X = mark#(X) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U71(mark(X1),X2)) mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X1 + X2 = active#(and(mark(X1),X2)) active#(isNatKind(0())) = 0 >= 0 = mark#(tt()) mark#(isNatIListKind(X)) = X >= X = active#(isNatIListKind(X)) active#(isNatKind(length(V1))) = V1 + 1 >= V1 = mark#(isNatIListKind(V1)) active#(isNatKind(s(V1))) = V1 >= V1 = mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) = L + N + 1 >= L + 1 = mark#(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),V1,V2)) = 0 >= 0 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 0 >= 0 = mark(U63(isNatIList(V2))) active(U63(tt())) = 0 >= 0 = mark(tt()) active(U71(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(U81(tt())) = 0 >= 0 = mark(nil()) active(U91(tt(),IL,M,N)) = IL + M + N >= IL + M + N = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = V1 + V2 >= V1 + V2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 + 1 >= V1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= V1 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 0 >= 0 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = IL >= 0 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = IL + M + N >= IL + M + N = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X1,X2)) = 0 >= 0 = active(U11(mark(X1),X2)) mark(tt()) = 0 >= 0 = active(tt()) mark(U12(X)) = 0 >= 0 = active(U12(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U21(X1,X2)) = 0 >= 0 = active(U21(mark(X1),X2)) mark(U22(X)) = X >= X = active(U22(mark(X))) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(U31(X1,X2)) = 0 >= 0 = active(U31(mark(X1),X2)) mark(U32(X)) = 0 >= 0 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 0 >= 0 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 0 >= 0 = active(U42(mark(X1),X2)) mark(U43(X)) = X >= X = active(U43(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 0 >= 0 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 0 >= 0 = active(U52(mark(X1),X2)) mark(U53(X)) = X >= X = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(U63(X)) = 0 >= 0 = active(U63(mark(X))) mark(U71(X1,X2)) = X2 + 1 >= X2 + 1 = active(U71(mark(X1),X2)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = active(length(mark(X))) mark(U81(X)) = 0 >= 0 = active(U81(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) mark(U91(X1,X2,X3,X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = X >= X = active(isNatIListKind(X)) mark(isNatKind(X)) = X >= X = active(isNatKind(X)) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = X >= X = U22(X) U22(active(X)) = X >= X = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = X >= X = U43(X) U43(active(X)) = X >= X = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = X >= X = U53(X) U53(active(X)) = X >= X = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(active(X1),X2) = X2 + 1 >= X2 + 1 = U71(X1,X2) U71(X1,active(X2)) = X2 + 1 >= X2 + 1 = U71(X1,X2) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = X2 + X3 + X4 >= X2 + X3 + X4 = U91(X1,X2,X3,X4) take(mark(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) and(mark(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,mark(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) and(active(X1),X2) = X1 + X2 >= X1 + X2 = and(X1,X2) and(X1,active(X2)) = X1 + X2 >= X1 + X2 = and(X1,X2) isNatIListKind(mark(X)) = X >= X = isNatIListKind(X) isNatIListKind(active(X)) = X >= X = isNatIListKind(X) isNatKind(mark(X)) = X >= X = isNatKind(X) isNatKind(active(X)) = X >= X = isNatKind(X) problem: DPs: mark#(isNatKind(X)) -> active#(isNatKind(X)) active#(U21(tt(),V1)) -> mark#(U22(isNat(V1))) active#(U22(tt())) -> mark#(tt()) mark#(isNatList(X)) -> active#(isNatList(X)) active#(U41(tt(),V1,V2)) -> mark#(U42(isNat(V1),V2)) mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2)) active#(U42(tt(),V2)) -> mark#(U43(isNatIList(V2))) mark#(U22(X)) -> mark#(X) mark#(U22(X)) -> active#(U22(mark(X))) active#(U43(tt())) -> mark#(tt()) mark#(isNat(X)) -> active#(isNat(X)) active#(U51(tt(),V1,V2)) -> mark#(U52(isNat(V1),V2)) active#(U52(tt(),V2)) -> mark#(U53(isNatList(V2))) active#(U53(tt())) -> mark#(tt()) mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3)) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U71(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(and(tt(),X)) -> mark#(X) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U71(X1,X2)) -> active#(U71(mark(X1),X2)) mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(s(V1))) -> mark#(isNatKind(V1)) active#(isNatList(cons(V1,V2))) -> mark#(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active#(length(cons(N,L))) -> mark#(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)), and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)), and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Open DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: cons#(mark(X1),X2) = X2 >= X2 = cons#(X1,X2) cons#(X1,mark(X2)) = X2 + 1 >= X2 = cons#(X1,X2) cons#(active(X1),X2) = X2 >= X2 = cons#(X1,X2) cons#(X1,active(X2)) = X2 + 1 >= X2 = cons#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: cons#(mark(X1),X2) = X1 + 1 >= X1 = cons#(X1,X2) cons#(active(X1),X2) = X1 + 1 >= X1 = cons#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatList#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: isNatList#(mark(X)) = X + 1 >= X = isNatList#(X) isNatList#(active(X)) = X + 1 >= X = isNatList#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U12#(mark(X)) -> U12#(X) U12#(active(X)) -> U12#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U12#(mark(X)) = X + 1 >= X = U12#(X) U12#(active(X)) = X + 1 >= X = U12#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: isNat#(mark(X)) = X + 1 >= X = isNat#(X) isNat#(active(X)) = X + 1 >= X = isNat#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U22#(mark(X)) -> U22#(X) U22#(active(X)) -> U22#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U22#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U22#(mark(X)) = X + 1 >= X = U22#(X) U22#(active(X)) = X + 1 >= X = U22#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U32#(mark(X)) -> U32#(X) U32#(active(X)) -> U32#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U32#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U32#(mark(X)) = X + 1 >= X = U32#(X) U32#(active(X)) = X + 1 >= X = U32#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U42#(mark(X1),X2) -> U42#(X1,X2) U42#(X1,mark(X2)) -> U42#(X1,X2) U42#(active(X1),X2) -> U42#(X1,X2) U42#(X1,active(X2)) -> U42#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U42#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U42#(mark(X1),X2) = X2 >= X2 = U42#(X1,X2) U42#(X1,mark(X2)) = X2 + 1 >= X2 = U42#(X1,X2) U42#(active(X1),X2) = X2 >= X2 = U42#(X1,X2) U42#(X1,active(X2)) = X2 + 1 >= X2 = U42#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U42#(mark(X1),X2) -> U42#(X1,X2) U42#(active(X1),X2) -> U42#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U42#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U42#(mark(X1),X2) = X1 + 1 >= X1 = U42#(X1,X2) U42#(active(X1),X2) = X1 + 1 >= X1 = U42#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatIList#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: isNatIList#(mark(X)) = X + 1 >= X = isNatIList#(X) isNatIList#(active(X)) = X + 1 >= X = isNatIList#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U43#(mark(X)) -> U43#(X) U43#(active(X)) -> U43#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U43#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U43#(mark(X)) = X + 1 >= X = U43#(X) U43#(active(X)) = X + 1 >= X = U43#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U52#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U52#(mark(X1),X2) = X2 >= X2 = U52#(X1,X2) U52#(X1,mark(X2)) = X2 + 1 >= X2 = U52#(X1,X2) U52#(active(X1),X2) = X2 >= X2 = U52#(X1,X2) U52#(X1,active(X2)) = X2 + 1 >= X2 = U52#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U52#(mark(X1),X2) -> U52#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U52#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U52#(mark(X1),X2) = X1 + 1 >= X1 = U52#(X1,X2) U52#(active(X1),X2) = X1 + 1 >= X1 = U52#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U53#(mark(X)) -> U53#(X) U53#(active(X)) -> U53#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U53#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U53#(mark(X)) = X + 1 >= X = U53#(X) U53#(active(X)) = X + 1 >= X = U53#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) U62#(active(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U62#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U62#(mark(X1),X2) = X2 >= X2 = U62#(X1,X2) U62#(X1,mark(X2)) = X2 + 1 >= X2 = U62#(X1,X2) U62#(active(X1),X2) = X2 >= X2 = U62#(X1,X2) U62#(X1,active(X2)) = X2 + 1 >= X2 = U62#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U62#(mark(X1),X2) -> U62#(X1,X2) U62#(active(X1),X2) -> U62#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U62#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U62#(mark(X1),X2) = X1 + 1 >= X1 = U62#(X1,X2) U62#(active(X1),X2) = X1 + 1 >= X1 = U62#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U63#(mark(X)) -> U63#(X) U63#(active(X)) -> U63#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U63#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U63#(mark(X)) = X + 1 >= X = U63#(X) U63#(active(X)) = X + 1 >= X = U63#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: length#(mark(X)) = X + 1 >= X = length#(X) length#(active(X)) = X + 1 >= X = length#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: s#(mark(X)) = X + 1 >= X = s#(X) s#(active(X)) = X + 1 >= X = s#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(X1,mark(X2)) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) take#(X1,active(X2)) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: take#(mark(X1),X2) = X2 >= X2 = take#(X1,X2) take#(X1,mark(X2)) = X2 + 1 >= X2 = take#(X1,X2) take#(active(X1),X2) = X2 >= X2 = take#(X1,X2) take#(X1,active(X2)) = X2 + 1 >= X2 = take#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: take#(mark(X1),X2) -> take#(X1,X2) take#(active(X1),X2) -> take#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [take#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: take#(mark(X1),X2) = X1 + 1 >= X1 = take#(X1,X2) take#(active(X1),X2) = X1 + 1 >= X1 = take#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: isNatIListKind#(mark(X)) -> isNatIListKind#(X) isNatIListKind#(active(X)) -> isNatIListKind#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatIListKind#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: isNatIListKind#(mark(X)) = X + 1 >= X = isNatIListKind#(X) isNatIListKind#(active(X)) = X + 1 >= X = isNatIListKind#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(X1,mark(X2)) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) U11#(X1,active(X2)) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U11#(mark(X1),X2) = X2 >= X2 = U11#(X1,X2) U11#(X1,mark(X2)) = X2 + 1 >= X2 = U11#(X1,X2) U11#(active(X1),X2) = X2 >= X2 = U11#(X1,X2) U11#(X1,active(X2)) = X2 + 1 >= X2 = U11#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U11#(mark(X1),X2) -> U11#(X1,X2) U11#(active(X1),X2) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U11#(mark(X1),X2) = X1 + 1 >= X1 = U11#(X1,X2) U11#(active(X1),X2) = X1 + 1 >= X1 = U11#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: isNatKind#(mark(X)) -> isNatKind#(X) isNatKind#(active(X)) -> isNatKind#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatKind#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: isNatKind#(mark(X)) = X + 1 >= X = isNatKind#(X) isNatKind#(active(X)) = X + 1 >= X = isNatKind#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U21#(mark(X1),X2) = X2 >= X2 = U21#(X1,X2) U21#(X1,mark(X2)) = X2 + 1 >= X2 = U21#(X1,X2) U21#(active(X1),X2) = X2 >= X2 = U21#(X1,X2) U21#(X1,active(X2)) = X2 + 1 >= X2 = U21#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U21#(mark(X1),X2) -> U21#(X1,X2) U21#(active(X1),X2) -> U21#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U21#(mark(X1),X2) = X1 + 1 >= X1 = U21#(X1,X2) U21#(active(X1),X2) = X1 + 1 >= X1 = U21#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U31#(mark(X1),X2) = X2 >= X2 = U31#(X1,X2) U31#(X1,mark(X2)) = X2 + 1 >= X2 = U31#(X1,X2) U31#(active(X1),X2) = X2 >= X2 = U31#(X1,X2) U31#(X1,active(X2)) = X2 + 1 >= X2 = U31#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U31#(mark(X1),X2) -> U31#(X1,X2) U31#(active(X1),X2) -> U31#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U31#(mark(X1),X2) = X1 + 1 >= X1 = U31#(X1,X2) U31#(active(X1),X2) = X1 + 1 >= X1 = U31#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(X1,mark(X2)) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) and#(X1,active(X2)) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: and#(mark(X1),X2) = X2 >= X2 = and#(X1,X2) and#(X1,mark(X2)) = X2 + 1 >= X2 = and#(X1,X2) and#(active(X1),X2) = X2 >= X2 = and#(X1,X2) and#(X1,active(X2)) = X2 + 1 >= X2 = and#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: and#(mark(X1),X2) -> and#(X1,X2) and#(active(X1),X2) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: and#(mark(X1),X2) = X1 + 1 >= X1 = and#(X1,X2) and#(active(X1),X2) = X1 + 1 >= X1 = and#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) U41#(active(X1),X2,X3) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1, x2) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = x1, [U51](x0, x1, x2) = x2, [U43](x0) = 0, [isNatIList](x0) = 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 1, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0 + 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U41#(mark(X1),X2,X3) = X1 + 1 >= X1 = U41#(X1,X2,X3) U41#(X1,mark(X2),X3) = X1 >= X1 = U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) = X1 >= X1 = U41#(X1,X2,X3) U41#(active(X1),X2,X3) = X1 + 1 >= X1 = U41#(X1,X2,X3) U41#(X1,active(X2),X3) = X1 >= X1 = U41#(X1,X2,X3) U41#(X1,X2,active(X3)) = X1 >= X1 = U41#(X1,X2,X3) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 2 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 + 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 2 >= 2 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 2 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 2 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 2 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 2 >= 2 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 2 >= 2 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 + 2 >= V2 + 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = X2 + 1 >= X2 + 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 2 >= X + 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 2 >= 2 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 2 >= 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X2 + 1 >= X2 + 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 2 >= 2 = active(isNatIListKind(X)) mark(isNatKind(X)) = 2 >= 2 = active(isNatKind(X)) cons(mark(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,mark(X2)) = X2 + 1 >= X2 = cons(X1,X2) cons(active(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,active(X2)) = X2 + 1 >= X2 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 2 >= X + 1 = isNatList(X) isNatList(active(X)) = X + 2 >= X + 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X3 + 1 >= X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X3 + 1 >= X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 + 1 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 + 1 >= X2 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) U41#(X1,active(X2),X3) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1, x2) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 0, [U12](x0) = 1, [isNatList](x0) = x0 + 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U41#(X1,mark(X2),X3) = X2 + 1 >= X2 = U41#(X1,X2,X3) U41#(X1,X2,mark(X3)) = X2 >= X2 = U41#(X1,X2,X3) U41#(X1,active(X2),X3) = X2 + 1 >= X2 = U41#(X1,X2,X3) U41#(X1,X2,active(X3)) = X2 >= X2 = U41#(X1,X2,X3) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 2 = mark(U12(isNatList(V1))) active(U12(tt())) = 2 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 2 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 2 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 2 >= 2 = active(U12(mark(X))) mark(isNatList(X)) = X + 2 >= X + 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = X + 2 >= X + 1 = isNatList(X) isNatList(active(X)) = X + 2 >= X + 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3) U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1, x2) = x2 + 1, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [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) = 0, [U43](x0) = 0, [isNatIList](x0) = 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U41#(X1,X2,mark(X3)) = X3 + 2 >= X3 + 1 = U41#(X1,X2,X3) U41#(X1,X2,active(X3)) = X3 + 2 >= X3 + 1 = U41#(X1,X2,X3) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 2 = mark(U12(isNatList(V1))) active(U12(tt())) = 2 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 2 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 2 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 2 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 2 >= 2 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 2 >= 2 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = X2 + 1 >= X2 + 1 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 2 >= 2 = active(U12(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 2 >= 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 2 >= 2 = active(isNatIListKind(X)) mark(isNatKind(X)) = 2 >= 2 = active(isNatKind(X)) cons(mark(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,mark(X2)) = X2 + 1 >= X2 = cons(X1,X2) cons(active(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,active(X2)) = X2 + 1 >= X2 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,mark(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) -> U51#(X1,X2,X3) U51#(active(X1),X2,X3) -> U51#(X1,X2,X3) U51#(X1,active(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,active(X3)) -> U51#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1, x2) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = x1, [U51](x0, x1, x2) = x2, [U43](x0) = 0, [isNatIList](x0) = 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 1, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0 + 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U51#(mark(X1),X2,X3) = X1 + 1 >= X1 = U51#(X1,X2,X3) U51#(X1,mark(X2),X3) = X1 >= X1 = U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) = X1 >= X1 = U51#(X1,X2,X3) U51#(active(X1),X2,X3) = X1 + 1 >= X1 = U51#(X1,X2,X3) U51#(X1,active(X2),X3) = X1 >= X1 = U51#(X1,X2,X3) U51#(X1,X2,active(X3)) = X1 >= X1 = U51#(X1,X2,X3) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 2 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 + 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 2 >= 2 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 2 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 2 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 2 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 2 >= 2 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 2 >= 2 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 + 2 >= V2 + 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = X2 + 1 >= X2 + 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 2 >= X + 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 2 >= 2 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 2 >= 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X2 + 1 >= X2 + 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 2 >= 2 = active(isNatIListKind(X)) mark(isNatKind(X)) = 2 >= 2 = active(isNatKind(X)) cons(mark(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,mark(X2)) = X2 + 1 >= X2 = cons(X1,X2) cons(active(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,active(X2)) = X2 + 1 >= X2 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 2 >= X + 1 = isNatList(X) isNatList(active(X)) = X + 2 >= X + 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X3 + 1 >= X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X3 + 1 >= X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 + 1 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 + 1 >= X2 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: U51#(X1,mark(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) -> U51#(X1,X2,X3) U51#(X1,active(X2),X3) -> U51#(X1,X2,X3) U51#(X1,X2,active(X3)) -> U51#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1, x2) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 0, [U12](x0) = 1, [isNatList](x0) = x0 + 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U51#(X1,mark(X2),X3) = X2 + 1 >= X2 = U51#(X1,X2,X3) U51#(X1,X2,mark(X3)) = X2 >= X2 = U51#(X1,X2,X3) U51#(X1,active(X2),X3) = X2 + 1 >= X2 = U51#(X1,X2,X3) U51#(X1,X2,active(X3)) = X2 >= X2 = U51#(X1,X2,X3) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 2 = mark(U12(isNatList(V1))) active(U12(tt())) = 2 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 2 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 2 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 2 >= 2 = active(U12(mark(X))) mark(isNatList(X)) = X + 2 >= X + 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = X + 2 >= X + 1 = isNatList(X) isNatList(active(X)) = X + 2 >= X + 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U51#(X1,X2,mark(X3)) -> U51#(X1,X2,X3) U51#(X1,X2,active(X3)) -> U51#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1, x2) = x2 + 1, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [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) = 0, [U43](x0) = 0, [isNatIList](x0) = 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U51#(X1,X2,mark(X3)) = X3 + 2 >= X3 + 1 = U51#(X1,X2,X3) U51#(X1,X2,active(X3)) = X3 + 2 >= X3 + 1 = U51#(X1,X2,X3) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 2 = mark(U12(isNatList(V1))) active(U12(tt())) = 2 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 2 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 2 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 2 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 2 >= 2 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 2 >= 2 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = X2 + 1 >= X2 + 1 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 2 >= 2 = active(U12(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 2 >= 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 2 >= 2 = active(isNatIListKind(X)) mark(isNatKind(X)) = 2 >= 2 = active(isNatKind(X)) cons(mark(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,mark(X2)) = X2 + 1 >= X2 = cons(X1,X2) cons(active(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,active(X2)) = X2 + 1 >= X2 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1, x2) = x0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [U53](x0) = 0, [U52](x0, x1) = x1, [U51](x0, x1, x2) = x2, [U43](x0) = 0, [isNatIList](x0) = 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 1, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0 + 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U61#(mark(X1),X2,X3) = X1 + 1 >= X1 = U61#(X1,X2,X3) U61#(X1,mark(X2),X3) = X1 >= X1 = U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) = X1 >= X1 = U61#(X1,X2,X3) U61#(active(X1),X2,X3) = X1 + 1 >= X1 = U61#(X1,X2,X3) U61#(X1,active(X2),X3) = X1 >= X1 = U61#(X1,X2,X3) U61#(X1,X2,active(X3)) = X1 >= X1 = U61#(X1,X2,X3) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 2 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = V2 + 1 >= V2 + 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 + 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 2 >= 2 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 2 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 2 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 2 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 2 >= 2 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 2 >= 2 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 + 2 >= V2 + 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = X2 + 1 >= X2 + 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 2 >= X + 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 2 >= 2 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 2 >= 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = X3 + 1 >= X3 + 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = X2 + 1 >= X2 + 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 2 >= 2 = active(isNatIListKind(X)) mark(isNatKind(X)) = 2 >= 2 = active(isNatKind(X)) cons(mark(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,mark(X2)) = X2 + 1 >= X2 = cons(X1,X2) cons(active(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,active(X2)) = X2 + 1 >= X2 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 2 >= X + 1 = isNatList(X) isNatList(active(X)) = X + 2 >= X + 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,mark(X2)) = 1 >= 1 = U31(X1,X2) U31(active(X1),X2) = 1 >= 1 = U31(X1,X2) U31(X1,active(X2)) = 1 >= 1 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = X3 + 1 >= X3 = U51(X1,X2,X3) U51(active(X1),X2,X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,active(X2),X3) = X3 >= X3 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = X3 + 1 >= X3 = U51(X1,X2,X3) U52(mark(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,mark(X2)) = X2 + 1 >= X2 = U52(X1,X2) U52(active(X1),X2) = X2 >= X2 = U52(X1,X2) U52(X1,active(X2)) = X2 + 1 >= X2 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1, x2) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](x0) = 1, [U71](x0, x1) = 0, [U63](x0) = 0, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 0, [U12](x0) = 1, [isNatList](x0) = x0 + 1, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U61#(X1,mark(X2),X3) = X2 + 1 >= X2 = U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) = X2 >= X2 = U61#(X1,X2,X3) U61#(X1,active(X2),X3) = X2 + 1 >= X2 = U61#(X1,X2,X3) U61#(X1,X2,active(X3)) = X2 >= X2 = U61#(X1,X2,X3) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 2 = mark(U12(isNatList(V1))) active(U12(tt())) = 2 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 2 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 2 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 2 >= 2 = active(U12(mark(X))) mark(isNatList(X)) = X + 2 >= X + 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = X + 2 >= X + 1 = isNatList(X) isNatList(active(X)) = X + 2 >= X + 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1, x2) = x2 + 1, [isNatKind](x0) = 1, [isNatIListKind](x0) = 1, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [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) = 0, [U43](x0) = 0, [isNatIList](x0) = 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0, [U21](x0, x1) = 1, [U12](x0) = 1, [isNatList](x0) = 0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U61#(X1,X2,mark(X3)) = X3 + 2 >= X3 + 1 = U61#(X1,X2,X3) U61#(X1,X2,active(X3)) = X3 + 2 >= X3 + 1 = U61#(X1,X2,X3) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 2 = mark(U12(isNatList(V1))) active(U12(tt())) = 2 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 2 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 2 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 2 >= 2 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 2 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 2 >= 2 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 2 >= 2 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = X2 + 1 >= X2 + 1 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 2 >= 2 = active(U12(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 1 >= X + 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 2 >= 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 2 >= 2 = active(isNatIListKind(X)) mark(isNatKind(X)) = 2 >= 2 = active(isNatKind(X)) cons(mark(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,mark(X2)) = X2 + 1 >= X2 = cons(X1,X2) cons(active(X1),X2) = X2 >= X2 = cons(X1,X2) cons(X1,active(X2)) = X2 + 1 >= X2 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 1 >= 1 = U12(X) U12(active(X)) = 1 >= 1 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 1 >= X = isNat(X) isNat(active(X)) = X + 1 >= X = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 1 >= 1 = isNatIListKind(X) isNatIListKind(active(X)) = 1 >= 1 = isNatIListKind(X) isNatKind(mark(X)) = 1 >= 1 = isNatKind(X) isNatKind(active(X)) = 1 >= 1 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U71#(mark(X1),X2) -> U71#(X1,X2) U71#(X1,mark(X2)) -> U71#(X1,X2) U71#(active(X1),X2) -> U71#(X1,X2) U71#(X1,active(X2)) -> U71#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U71#](x0, x1) = x1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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) = 1, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 1, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U71#(mark(X1),X2) = X2 >= X2 = U71#(X1,X2) U71#(X1,mark(X2)) = X2 + 1 >= X2 = U71#(X1,X2) U71#(active(X1),X2) = X2 >= X2 = U71#(X1,X2) U71#(X1,active(X2)) = X2 + 1 >= X2 = U71#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 2 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 2 >= 2 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 2 >= 2 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 1 >= 1 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 1 >= 1 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 1 >= 1 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U71#(mark(X1),X2) -> U71#(X1,X2) U71#(active(X1),X2) -> U71#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U71#](x0, x1) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 1, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [U71](x0, x1) = 1, [U63](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 1, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U71#(mark(X1),X2) = X1 + 1 >= X1 = U71#(X1,X2) U71#(active(X1),X2) = X1 + 1 >= X1 = U71#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 2 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 2 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 2 >= 2 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 3 >= 2 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 2 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 2 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 2 >= 2 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 1 >= X + 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 2 >= 2 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,mark(X2)) = 1 >= 1 = U21(X1,X2) U21(active(X1),X2) = 1 >= 1 = U21(X1,X2) U21(X1,active(X2)) = 1 >= 1 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 1 >= X = isNatIList(X) isNatIList(active(X)) = X + 1 >= X = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 1 >= 1 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 1 >= 1 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U81#(mark(X)) -> U81#(X) U81#(active(X)) -> U81#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U81#](x0) = x0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 1, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 1, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = x0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U81#(mark(X)) = X + 1 >= X = U81#(X) U81#(active(X)) = X + 1 >= X = U81#(X) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 2 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 2 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 2 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 2 >= 2 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 2 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = X + 1 >= X + 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(U81(X)) = 2 >= 2 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 2 >= 2 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = X + 1 >= X = isNatList(X) isNatList(active(X)) = X + 1 >= X = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) U81(mark(X)) = 1 >= 1 = U81(X) U81(active(X)) = 1 >= 1 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,mark(X2)) = 1 >= 1 = take(X1,X2) take(active(X1),X2) = 1 >= 1 = take(X1,X2) take(X1,active(X2)) = 1 >= 1 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed DPs: U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,mark(X2),X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,mark(X3),X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,X3,mark(X4)) -> U91#(X1,X2,X3,X4) U91#(active(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,active(X2),X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,active(X3),X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,X3,active(X4)) -> U91#(X1,X2,X3,X4) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U91#](x0, x1, x2, x3) = x2, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 0, [length](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, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = 1, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U91#(mark(X1),X2,X3,X4) = X3 >= X3 = U91#(X1,X2,X3,X4) U91#(X1,mark(X2),X3,X4) = X3 >= X3 = U91#(X1,X2,X3,X4) U91#(X1,X2,mark(X3),X4) = X3 + 1 >= X3 = U91#(X1,X2,X3,X4) U91#(X1,X2,X3,mark(X4)) = X3 >= X3 = U91#(X1,X2,X3,X4) U91#(active(X1),X2,X3,X4) = X3 >= X3 = U91#(X1,X2,X3,X4) U91#(X1,active(X2),X3,X4) = X3 >= X3 = U91#(X1,X2,X3,X4) U91#(X1,X2,active(X3),X4) = X3 + 1 >= X3 = U91#(X1,X2,X3,X4) U91#(X1,X2,X3,active(X4)) = X3 >= X3 = U91#(X1,X2,X3,X4) active(zeros()) = 2 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 3 >= 1 = mark(tt()) active(isNat(length(V1))) = 3 >= 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X1,X2)) = 2 >= 2 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = X + 2 >= X + 2 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 1 >= 1 = active(U71(mark(X1),X2)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,mark(X2)) = 1 >= 1 = U11(X1,X2) U11(active(X1),X2) = 1 >= 1 = U11(X1,X2) U11(X1,active(X2)) = 1 >= 1 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = X + 2 >= X + 1 = isNat(X) isNat(active(X)) = X + 2 >= X + 1 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,mark(X2)) = 0 >= 0 = U71(X1,X2) U71(active(X1),X2) = 0 >= 0 = U71(X1,X2) U71(X1,active(X2)) = 0 >= 0 = U71(X1,X2) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,mark(X2),X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,X3,mark(X4)) -> U91#(X1,X2,X3,X4) U91#(active(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,active(X2),X3,X4) -> U91#(X1,X2,X3,X4) U91#(X1,X2,X3,active(X4)) -> U91#(X1,X2,X3,X4) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U91#](x0, x1, x2, x3) = x0 + x1 + x3 + 1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x1, [take](x0, x1) = 0, [U91](x0, x1, x2, x3) = 0, [nil] = 0, [U81](x0) = 0, [s](x0) = 1, [length](x0) = 1, [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) = 0, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = 0, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 1, [active](x0) = x0 + 1, [zeros] = 0 orientation: U91#(mark(X1),X2,X3,X4) = X1 + X2 + X4 + 2 >= X1 + X2 + X4 + 1 = U91#(X1,X2,X3,X4) U91#(X1,mark(X2),X3,X4) = X1 + X2 + X4 + 2 >= X1 + X2 + X4 + 1 = U91#(X1,X2,X3,X4) U91#(X1,X2,X3,mark(X4)) = X1 + X2 + X4 + 2 >= X1 + X2 + X4 + 1 = U91#(X1,X2,X3,X4) U91#(active(X1),X2,X3,X4) = X1 + X2 + X4 + 2 >= X1 + X2 + X4 + 1 = U91#(X1,X2,X3,X4) U91#(X1,active(X2),X3,X4) = X1 + X2 + X4 + 2 >= X1 + X2 + X4 + 1 = U91#(X1,X2,X3,X4) U91#(X1,X2,X3,active(X4)) = X1 + X2 + X4 + 2 >= X1 + X2 + X4 + 1 = U91#(X1,X2,X3,X4) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),V1,V2)) = 1 >= 1 = mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) = 1 >= 1 = mark(U63(isNatIList(V2))) active(U63(tt())) = 1 >= 1 = mark(tt()) active(U71(tt(),L)) = 2 >= 2 = mark(s(length(L))) active(U81(tt())) = 1 >= 1 = mark(nil()) active(U91(tt(),IL,M,N)) = 1 >= 1 = mark(cons(N,take(M,IL))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) = 1 >= 1 = mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(take(0(),IL)) = 1 >= 1 = mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) = 1 >= 1 = mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M), isNatKind (M)),and( isNat(N), isNatKind ( N)))), IL,M,N)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X1,X2)) = 1 >= 1 = active(U11(mark(X1),X2)) mark(tt()) = 1 >= 1 = active(tt()) mark(U12(X)) = 1 >= 1 = active(U12(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U21(X1,X2)) = 1 >= 1 = active(U21(mark(X1),X2)) mark(U22(X)) = 1 >= 1 = active(U22(mark(X))) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(U31(X1,X2)) = 1 >= 1 = active(U31(mark(X1),X2)) mark(U32(X)) = 1 >= 1 = active(U32(mark(X))) mark(U41(X1,X2,X3)) = 1 >= 1 = active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) = 1 >= 1 = active(U42(mark(X1),X2)) mark(U43(X)) = 1 >= 1 = active(U43(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2,X3)) = 1 >= 1 = active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) = 1 >= 1 = active(U52(mark(X1),X2)) mark(U53(X)) = 1 >= 1 = active(U53(mark(X))) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(U63(X)) = 1 >= 1 = active(U63(mark(X))) mark(U71(X1,X2)) = 2 >= 2 = active(U71(mark(X1),X2)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(U81(X)) = 1 >= 1 = active(U81(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) mark(U91(X1,X2,X3,X4)) = 1 >= 1 = active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) = 1 >= 1 = active(take(mark(X1),mark(X2))) mark(and(X1,X2)) = X2 + 1 >= X2 + 1 = active(and(mark(X1),X2)) mark(isNatIListKind(X)) = 1 >= 1 = active(isNatIListKind(X)) mark(isNatKind(X)) = 1 >= 1 = active(isNatKind(X)) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,mark(X2)) = 0 >= 0 = U11(X1,X2) U11(active(X1),X2) = 0 >= 0 = U11(X1,X2) U11(X1,active(X2)) = 0 >= 0 = U11(X1,X2) U12(mark(X)) = 0 >= 0 = U12(X) U12(active(X)) = 0 >= 0 = U12(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U21(mark(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,mark(X2)) = 0 >= 0 = U21(X1,X2) U21(active(X1),X2) = 0 >= 0 = U21(X1,X2) U21(X1,active(X2)) = 0 >= 0 = U21(X1,X2) U22(mark(X)) = 0 >= 0 = U22(X) U22(active(X)) = 0 >= 0 = U22(X) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) U31(mark(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,mark(X2)) = 0 >= 0 = U31(X1,X2) U31(active(X1),X2) = 0 >= 0 = U31(X1,X2) U31(X1,active(X2)) = 0 >= 0 = U31(X1,X2) U32(mark(X)) = 0 >= 0 = U32(X) U32(active(X)) = 0 >= 0 = U32(X) U41(mark(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,mark(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,mark(X3)) = 0 >= 0 = U41(X1,X2,X3) U41(active(X1),X2,X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,active(X2),X3) = 0 >= 0 = U41(X1,X2,X3) U41(X1,X2,active(X3)) = 0 >= 0 = U41(X1,X2,X3) U42(mark(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,mark(X2)) = 0 >= 0 = U42(X1,X2) U42(active(X1),X2) = 0 >= 0 = U42(X1,X2) U42(X1,active(X2)) = 0 >= 0 = U42(X1,X2) U43(mark(X)) = 0 >= 0 = U43(X) U43(active(X)) = 0 >= 0 = U43(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,mark(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,mark(X3)) = 0 >= 0 = U51(X1,X2,X3) U51(active(X1),X2,X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,active(X2),X3) = 0 >= 0 = U51(X1,X2,X3) U51(X1,X2,active(X3)) = 0 >= 0 = U51(X1,X2,X3) U52(mark(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,mark(X2)) = 0 >= 0 = U52(X1,X2) U52(active(X1),X2) = 0 >= 0 = U52(X1,X2) U52(X1,active(X2)) = 0 >= 0 = U52(X1,X2) U53(mark(X)) = 0 >= 0 = U53(X) U53(active(X)) = 0 >= 0 = U53(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) U63(mark(X)) = 0 >= 0 = U63(X) U63(active(X)) = 0 >= 0 = U63(X) U71(mark(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,mark(X2)) = 1 >= 1 = U71(X1,X2) U71(active(X1),X2) = 1 >= 1 = U71(X1,X2) U71(X1,active(X2)) = 1 >= 1 = U71(X1,X2) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) U81(mark(X)) = 0 >= 0 = U81(X) U81(active(X)) = 0 >= 0 = U81(X) U91(mark(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) = 0 >= 0 = U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) = 0 >= 0 = U91(X1,X2,X3,X4) take(mark(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,mark(X2)) = 0 >= 0 = take(X1,X2) take(active(X1),X2) = 0 >= 0 = take(X1,X2) take(X1,active(X2)) = 0 >= 0 = take(X1,X2) and(mark(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,mark(X2)) = X2 + 1 >= X2 = and(X1,X2) and(active(X1),X2) = X2 >= X2 = and(X1,X2) and(X1,active(X2)) = X2 + 1 >= X2 = and(X1,X2) isNatIListKind(mark(X)) = 0 >= 0 = isNatIListKind(X) isNatIListKind(active(X)) = 0 >= 0 = isNatIListKind(X) isNatKind(mark(X)) = 0 >= 0 = isNatKind(X) isNatKind(active(X)) = 0 >= 0 = isNatKind(X) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),V1,V2)) -> mark(U62(isNat(V1),V2)) active(U62(tt(),V2)) -> mark(U63(isNatIList(V2))) active(U63(tt())) -> mark(tt()) active(U71(tt(),L)) -> mark(s(length(L))) active(U81(tt())) -> mark(nil()) active(U91(tt(),IL,M,N)) -> mark(cons(N,take(M,IL))) active(and(tt(),X)) -> mark(X) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) -> mark(tt()) active(isNatIListKind(zeros())) -> mark(tt()) active(isNatIListKind(cons(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatIListKind(take(V1,V2))) -> mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) -> mark(tt()) active(isNatKind(length(V1))) -> mark(isNatIListKind(V1)) active(isNatKind(s(V1))) -> mark(isNatKind(V1)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatList(take(V1,V2))) -> mark(U61(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U71(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) active(take(0(),IL)) -> mark(U81(and(isNatIList(IL),isNatIListKind(IL)))) active(take(s(M),cons(N,IL))) -> mark(U91(and(and(isNatIList(IL),isNatIListKind(IL)),and(and(isNat(M),isNatKind(M)), and(isNat(N),isNatKind(N)))), IL,M,N)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X1,X2)) -> active(U11(mark(X1),X2)) mark(tt()) -> active(tt()) mark(U12(X)) -> active(U12(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X)) -> active(U22(mark(X))) mark(isNat(X)) -> active(isNat(X)) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3)) mark(U42(X1,X2)) -> active(U42(mark(X1),X2)) mark(U43(X)) -> active(U43(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2,X3)) -> active(U51(mark(X1),X2,X3)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U53(X)) -> active(U53(mark(X))) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(U63(X)) -> active(U63(mark(X))) mark(U71(X1,X2)) -> active(U71(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil()) -> active(nil()) mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X1),X2) -> U11(X1,X2) U11(X1,mark(X2)) -> U11(X1,X2) U11(active(X1),X2) -> U11(X1,X2) U11(X1,active(X2)) -> U11(X1,X2) U12(mark(X)) -> U12(X) U12(active(X)) -> U12(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U21(active(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U22(mark(X)) -> U22(X) U22(active(X)) -> U22(X) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U31(active(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U32(mark(X)) -> U32(X) U32(active(X)) -> U32(X) U41(mark(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,mark(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,mark(X3)) -> U41(X1,X2,X3) U41(active(X1),X2,X3) -> U41(X1,X2,X3) U41(X1,active(X2),X3) -> U41(X1,X2,X3) U41(X1,X2,active(X3)) -> U41(X1,X2,X3) U42(mark(X1),X2) -> U42(X1,X2) U42(X1,mark(X2)) -> U42(X1,X2) U42(active(X1),X2) -> U42(X1,X2) U42(X1,active(X2)) -> U42(X1,X2) U43(mark(X)) -> U43(X) U43(active(X)) -> U43(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,mark(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,mark(X3)) -> U51(X1,X2,X3) U51(active(X1),X2,X3) -> U51(X1,X2,X3) U51(X1,active(X2),X3) -> U51(X1,X2,X3) U51(X1,X2,active(X3)) -> U51(X1,X2,X3) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U53(mark(X)) -> U53(X) U53(active(X)) -> U53(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) U63(mark(X)) -> U63(X) U63(active(X)) -> U63(X) U71(mark(X1),X2) -> U71(X1,X2) U71(X1,mark(X2)) -> U71(X1,X2) U71(active(X1),X2) -> U71(X1,X2) U71(X1,active(X2)) -> U71(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4) U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4) U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4) U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4) U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isNatIListKind(mark(X)) -> isNatIListKind(X) isNatIListKind(active(X)) -> isNatIListKind(X) isNatKind(mark(X)) -> isNatKind(X) isNatKind(active(X)) -> isNatKind(X) Qed