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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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(),L)) -> length#(L) active#(U61(tt(),L)) -> s#(length(L)) active#(U61(tt(),L)) -> mark#(s(length(L))) 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#(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#(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))) -> U61#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(length(cons(N,L))) -> mark#(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> mark#(X1) mark#(U61(X1,X2)) -> U61#(mark(X1),X2) mark#(U61(X1,X2)) -> active#(U61(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#(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)) mark#(nil()) -> active#(nil()) 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) -> U61#(X1,X2) U61#(X1,mark(X2)) -> U61#(X1,X2) U61#(active(X1),X2) -> U61#(X1,X2) U61#(X1,active(X2)) -> U61#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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(),L)) -> length#(L) active#(U61(tt(),L)) -> s#(length(L)) active#(U61(tt(),L)) -> mark#(s(length(L))) 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#(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#(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))) -> U61#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(length(cons(N,L))) -> mark#(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> mark#(X1) mark#(U61(X1,X2)) -> U61#(mark(X1),X2) mark#(U61(X1,X2)) -> active#(U61(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#(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)) mark#(nil()) -> active#(nil()) 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) -> U61#(X1,X2) U61#(X1,mark(X2)) -> U61#(X1,X2) U61#(active(X1),X2) -> U61#(X1,X2) U61#(X1,active(X2)) -> U61#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: ... Restore Modifier: 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(),L)) -> length#(L) active#(U61(tt(),L)) -> s#(length(L)) active#(U61(tt(),L)) -> mark#(s(length(L))) 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#(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#(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))) -> U61#(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) active#(length(cons(N,L))) -> mark#(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> mark#(X1) mark#(U61(X1,X2)) -> U61#(mark(X1),X2) mark#(U61(X1,X2)) -> active#(U61(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#(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)) mark#(nil()) -> active#(nil()) 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) -> U61#(X1,X2) U61#(X1,mark(X2)) -> U61#(X1,X2) U61#(active(X1),X2) -> U61#(X1,X2) U61#(X1,active(X2)) -> U61#(X1,X2) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(X)) -> length#(X) 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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) SCC Processor: #sccs: 23 #rules: 142 #arcs: 5258/41209 DPs: mark#(nil()) -> active#(nil()) 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(),L)) -> mark#(s(length(L))) mark#(U42(X1,X2)) -> mark#(X1) mark#(U42(X1,X2)) -> active#(U42(mark(X1),X2)) active#(and(tt(),X)) -> mark#(X) mark#(U43(X)) -> mark#(X) mark#(U43(X)) -> active#(U43(mark(X))) active#(isNat(0())) -> mark#(tt()) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1)) mark#(U51(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2,X3)) -> active#(U51(mark(X1),X2,X3)) active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1)) mark#(U52(X1,X2)) -> mark#(X1) mark#(U52(X1,X2)) -> active#(U52(mark(X1),X2)) active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V)) mark#(U53(X)) -> mark#(X) mark#(U53(X)) -> active#(U53(mark(X))) active#(isNatIList(zeros())) -> mark#(tt()) mark#(U61(X1,X2)) -> mark#(X1) mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatIListKind(nil())) -> mark#(tt()) mark#(length(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(isNatIListKind(zeros())) -> mark#(tt()) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) active#(isNatIListKind(cons(V1,V2))) -> mark#(and(isNatKind(V1),isNatIListKind(V2))) mark#(isNatIListKind(X)) -> active#(isNatIListKind(X)) active#(isNatKind(0())) -> mark#(tt()) mark#(isNatKind(X)) -> active#(isNatKind(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(nil())) -> mark#(0()) active#(length(cons(N,L))) -> mark#(U61(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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: 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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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: U61#(mark(X1),X2) -> U61#(X1,X2) U61#(X1,mark(X2)) -> U61#(X1,X2) U61#(active(X1),X2) -> U61#(X1,X2) U61#(X1,active(X2)) -> U61#(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(),L)) -> mark(s(length(L))) 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(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(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L)) 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)) -> active(U61(mark(X1),X2)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(isNatIListKind(X)) -> active(isNatIListKind(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(nil()) -> active(nil()) 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) -> U61(X1,X2) U61(X1,mark(X2)) -> U61(X1,X2) U61(active(X1),X2) -> U61(X1,X2) U61(X1,active(X2)) -> U61(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) 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