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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),V1)) -> isNatList#(V1) active#(U11(tt(),V1)) -> U12#(isNatList(V1)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U31(tt(),V)) -> isNatList#(V) active#(U31(tt(),V)) -> U32#(isNatList(V)) active#(U41(tt(),V1,V2)) -> isNat#(V1) active#(U41(tt(),V1,V2)) -> U42#(isNat(V1),V2) active#(U42(tt(),V2)) -> isNatIList#(V2) active#(U42(tt(),V2)) -> U43#(isNatIList(V2)) active#(U51(tt(),V1,V2)) -> isNat#(V1) active#(U51(tt(),V1,V2)) -> U52#(isNat(V1),V2) active#(U52(tt(),V2)) -> isNatList#(V2) active#(U52(tt(),V2)) -> U53#(isNatList(V2)) active#(U61(tt(),L)) -> length#(L) active#(U61(tt(),L)) -> s#(length(L)) active#(isNat(length(V1))) -> isNatIListKind#(V1) active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatIList(V)) -> isNatIListKind#(V) active#(isNatIList(V)) -> U31#(isNatIListKind(V),V) active#(isNatIList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIList(cons(V1,V2))) -> U41#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatIListKind(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatKind(length(V1))) -> isNatIListKind#(V1) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(cons(V1,V2))) -> U51#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(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#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X)) -> active#(X) active#(U12(X)) -> U12#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U32(X)) -> active#(X) active#(U32(X)) -> U32#(active(X)) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(U42(X1,X2)) -> active#(X1) active#(U42(X1,X2)) -> U42#(active(X1),X2) active#(U43(X)) -> active#(X) active#(U43(X)) -> U43#(active(X)) active#(U51(X1,X2,X3)) -> active#(X1) active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3) active#(U52(X1,X2)) -> active#(X1) active#(U52(X1,X2)) -> U52#(active(X1),X2) active#(U53(X)) -> active#(X) active#(U53(X)) -> U53#(active(X)) active#(U61(X1,X2)) -> active#(X1) active#(U61(X1,X2)) -> U61#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U42#(mark(X1),X2) -> U42#(X1,X2) U43#(mark(X)) -> U43#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U52#(mark(X1),X2) -> U52#(X1,X2) U53#(mark(X)) -> U53#(X) U61#(mark(X1),X2) -> U61#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) and#(mark(X1),X2) -> and#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X)) -> proper#(X) proper#(U12(X)) -> U12#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U32(X)) -> proper#(X) proper#(U32(X)) -> U32#(proper(X)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(U42(X1,X2)) -> proper#(X2) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2)) proper#(U43(X)) -> proper#(X) proper#(U43(X)) -> U43#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3)) proper#(U52(X1,X2)) -> proper#(X2) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2)) proper#(U53(X)) -> proper#(X) proper#(U53(X)) -> U53#(proper(X)) proper#(U61(X1,X2)) -> proper#(X2) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatIListKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> isNatIListKind#(proper(X)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U12#(ok(X)) -> U12#(X) isNatList#(ok(X)) -> isNatList#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) isNat#(ok(X)) -> isNat#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U32#(ok(X)) -> U32#(X) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U42#(ok(X1),ok(X2)) -> U42#(X1,X2) U43#(ok(X)) -> U43#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) U52#(ok(X1),ok(X2)) -> U52#(X1,X2) U53#(ok(X)) -> U53#(X) U61#(ok(X1),ok(X2)) -> U61#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatIListKind#(ok(X)) -> isNatIListKind#(X) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) EDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U11(tt(),V1)) -> isNatList#(V1) active#(U11(tt(),V1)) -> U12#(isNatList(V1)) active#(U21(tt(),V1)) -> isNat#(V1) active#(U21(tt(),V1)) -> U22#(isNat(V1)) active#(U31(tt(),V)) -> isNatList#(V) active#(U31(tt(),V)) -> U32#(isNatList(V)) active#(U41(tt(),V1,V2)) -> isNat#(V1) active#(U41(tt(),V1,V2)) -> U42#(isNat(V1),V2) active#(U42(tt(),V2)) -> isNatIList#(V2) active#(U42(tt(),V2)) -> U43#(isNatIList(V2)) active#(U51(tt(),V1,V2)) -> isNat#(V1) active#(U51(tt(),V1,V2)) -> U52#(isNat(V1),V2) active#(U52(tt(),V2)) -> isNatList#(V2) active#(U52(tt(),V2)) -> U53#(isNatList(V2)) active#(U61(tt(),L)) -> length#(L) active#(U61(tt(),L)) -> s#(length(L)) active#(isNat(length(V1))) -> isNatIListKind#(V1) active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1) active#(isNat(s(V1))) -> isNatKind#(V1) active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1) active#(isNatIList(V)) -> isNatIListKind#(V) active#(isNatIList(V)) -> U31#(isNatIListKind(V),V) active#(isNatIList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatIList(cons(V1,V2))) -> U41#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(isNatIListKind(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1) active#(isNatIListKind(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatKind(length(V1))) -> isNatIListKind#(V1) active#(isNatKind(s(V1))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> isNatIListKind#(V2) active#(isNatList(cons(V1,V2))) -> isNatKind#(V1) active#(isNatList(cons(V1,V2))) -> and#(isNatKind(V1),isNatIListKind(V2)) active#(isNatList(cons(V1,V2))) -> U51#(and(isNatKind(V1),isNatIListKind(V2)),V1,V2) active#(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#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X1,X2)) -> active#(X1) active#(U11(X1,X2)) -> U11#(active(X1),X2) active#(U12(X)) -> active#(X) active#(U12(X)) -> U12#(active(X)) active#(U21(X1,X2)) -> active#(X1) active#(U21(X1,X2)) -> U21#(active(X1),X2) active#(U22(X)) -> active#(X) active#(U22(X)) -> U22#(active(X)) active#(U31(X1,X2)) -> active#(X1) active#(U31(X1,X2)) -> U31#(active(X1),X2) active#(U32(X)) -> active#(X) active#(U32(X)) -> U32#(active(X)) active#(U41(X1,X2,X3)) -> active#(X1) active#(U41(X1,X2,X3)) -> U41#(active(X1),X2,X3) active#(U42(X1,X2)) -> active#(X1) active#(U42(X1,X2)) -> U42#(active(X1),X2) active#(U43(X)) -> active#(X) active#(U43(X)) -> U43#(active(X)) active#(U51(X1,X2,X3)) -> active#(X1) active#(U51(X1,X2,X3)) -> U51#(active(X1),X2,X3) active#(U52(X1,X2)) -> active#(X1) active#(U52(X1,X2)) -> U52#(active(X1),X2) active#(U53(X)) -> active#(X) active#(U53(X)) -> U53#(active(X)) active#(U61(X1,X2)) -> active#(X1) active#(U61(X1,X2)) -> U61#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) active#(and(X1,X2)) -> active#(X1) active#(and(X1,X2)) -> and#(active(X1),X2) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) U12#(mark(X)) -> U12#(X) U21#(mark(X1),X2) -> U21#(X1,X2) U22#(mark(X)) -> U22#(X) U31#(mark(X1),X2) -> U31#(X1,X2) U32#(mark(X)) -> U32#(X) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) U42#(mark(X1),X2) -> U42#(X1,X2) U43#(mark(X)) -> U43#(X) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) U52#(mark(X1),X2) -> U52#(X1,X2) U53#(mark(X)) -> U53#(X) U61#(mark(X1),X2) -> U61#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) and#(mark(X1),X2) -> and#(X1,X2) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> U11#(proper(X1),proper(X2)) proper#(U12(X)) -> proper#(X) proper#(U12(X)) -> U12#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U21(X1,X2)) -> proper#(X2) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> U21#(proper(X1),proper(X2)) proper#(U22(X)) -> proper#(X) proper#(U22(X)) -> U22#(proper(X)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(U31(X1,X2)) -> proper#(X2) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> U31#(proper(X1),proper(X2)) proper#(U32(X)) -> proper#(X) proper#(U32(X)) -> U32#(proper(X)) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> U41#(proper(X1),proper(X2),proper(X3)) proper#(U42(X1,X2)) -> proper#(X2) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> U42#(proper(X1),proper(X2)) proper#(U43(X)) -> proper#(X) proper#(U43(X)) -> U43#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> U51#(proper(X1),proper(X2),proper(X3)) proper#(U52(X1,X2)) -> proper#(X2) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> U52#(proper(X1),proper(X2)) proper#(U53(X)) -> proper#(X) proper#(U53(X)) -> U53#(proper(X)) proper#(U61(X1,X2)) -> proper#(X2) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> U61#(proper(X1),proper(X2)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) proper#(and(X1,X2)) -> proper#(X2) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) proper#(isNatIListKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> isNatIListKind#(proper(X)) proper#(isNatKind(X)) -> proper#(X) proper#(isNatKind(X)) -> isNatKind#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U12#(ok(X)) -> U12#(X) isNatList#(ok(X)) -> isNatList#(X) U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U22#(ok(X)) -> U22#(X) isNat#(ok(X)) -> isNat#(X) U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U32#(ok(X)) -> U32#(X) U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U42#(ok(X1),ok(X2)) -> U42#(X1,X2) U43#(ok(X)) -> U43#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) U52#(ok(X1),ok(X2)) -> U52#(X1,X2) U53#(ok(X)) -> U53#(X) U61#(ok(X1),ok(X2)) -> U61#(X1,X2) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(X) and#(ok(X1),ok(X2)) -> and#(X1,X2) isNatIListKind#(ok(X)) -> isNatIListKind#(X) isNatKind#(ok(X)) -> isNatKind#(X) top#(mark(X)) -> proper#(X) top#(mark(X)) -> top#(proper(X)) top#(ok(X)) -> active#(X) top#(ok(X)) -> top#(active(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 25 #rules: 92 #arcs: 3572/31329 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: proper#(isNatKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> proper#(X2) proper#(U53(X)) -> proper#(X) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(isNatIList(X)) -> proper#(X) proper#(U43(X)) -> proper#(X) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(isNat(X)) -> proper#(X) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(isNatList(X)) -> proper#(X) proper#(U12(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: proper#(isNatKind(X)) = X >= X = proper#(X) proper#(isNatIListKind(X)) = X >= X = proper#(X) proper#(and(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(length(X)) = X >= X = proper#(X) proper#(s(X)) = X + 1 >= X = proper#(X) proper#(U61(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U61(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U53(X)) = X >= X = proper#(X) proper#(U52(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U52(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = proper#(X1) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X2 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X3 = proper#(X3) proper#(isNatIList(X)) = X >= X = proper#(X) proper#(U43(X)) = X >= X = proper#(X) proper#(U42(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U42(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X2 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X3 = proper#(X3) proper#(U32(X)) = X >= X = proper#(X) proper#(U31(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(isNat(X)) = X >= X = proper#(X) proper#(U22(X)) = X >= X = proper#(X) proper#(U21(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(isNatList(X)) = X >= X = proper#(X) proper#(U12(X)) = X >= X = proper#(X) proper#(U11(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 >= X2 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 + 1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),V)) = V + 1 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 + 1 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 + 1 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 + 1 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 + 1 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 0 = mark(tt()) active(U61(tt(),L)) = L + 1 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 + V2 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 + 1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + X2 >= X1 + X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + X2 >= X1 + X2 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 + X2 >= X1 + X2 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X2 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = X >= X = U12(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + X2 >= X1 + X2 = U42(proper(X1),proper(X2)) proper(U43(X)) = X >= X = U43(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + X2 >= X1 + X2 = U52(proper(X1),proper(X2)) proper(U53(X)) = X >= X = U53(proper(X)) proper(U61(X1,X2)) = X1 + X2 >= X1 + X2 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X >= X = isNatIListKind(proper(X)) proper(isNatKind(X)) = X >= X = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = X >= X = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X >= X = ok(isNatIListKind(X)) isNatKind(ok(X)) = X >= X = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(isNatKind(X)) -> proper#(X) proper#(isNatIListKind(X)) -> proper#(X) proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> proper#(X2) proper#(U53(X)) -> proper#(X) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(isNatIList(X)) -> proper#(X) proper#(U43(X)) -> proper#(X) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(isNat(X)) -> proper#(X) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(isNatList(X)) -> proper#(X) proper#(U12(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = x0 + 1, [isNatIListKind](x0) = x0 + 1, [and](x0, x1) = x0 + x1, [s](x0) = 0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + x1 + 1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: proper#(isNatKind(X)) = X + 2 >= X + 1 = proper#(X) proper#(isNatIListKind(X)) = X + 2 >= X + 1 = proper#(X) proper#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(length(X)) = X + 1 >= X + 1 = proper#(X) proper#(U61(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U61(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U53(X)) = X + 1 >= X + 1 = proper#(X) proper#(U52(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U52(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + 1 = proper#(X1) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X2 + 1 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X3 + 1 = proper#(X3) proper#(isNatIList(X)) = X + 1 >= X + 1 = proper#(X) proper#(U43(X)) = X + 1 >= X + 1 = proper#(X) proper#(U42(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U42(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + 1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X2 + 1 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X3 + 1 = proper#(X3) proper#(U32(X)) = X + 1 >= X + 1 = proper#(X) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(isNat(X)) = X + 1 >= X + 1 = proper#(X) proper#(U22(X)) = X + 1 >= X + 1 = proper#(X) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(isNatList(X)) = X + 1 >= X + 1 = proper#(X) proper#(U12(X)) = X + 1 >= X + 1 = proper#(X) proper#(U11(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = L >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 + 1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 + V2 + 2 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 + 1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 + 1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + X2 >= X1 + X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + X2 >= X1 + X2 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 + X2 >= X1 + X2 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 + 1 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X2 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = X >= X = U12(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + X2 >= X1 + X2 = U42(proper(X1),proper(X2)) proper(U43(X)) = X >= X = U43(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + X2 >= X1 + X2 = U52(proper(X1),proper(X2)) proper(U53(X)) = X >= X = U53(proper(X)) proper(U61(X1,X2)) = X1 + X2 >= X1 + X2 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = X >= X = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = X >= X = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> proper#(X2) proper#(U53(X)) -> proper#(X) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(isNatIList(X)) -> proper#(X) proper#(U43(X)) -> proper#(X) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(isNat(X)) -> proper#(X) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(isNatList(X)) -> proper#(X) proper#(U12(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = 1, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [isNat](x0) = x0 + 1, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: proper#(and(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(length(X)) = X >= X = proper#(X) proper#(U61(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U61(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U53(X)) = X >= X = proper#(X) proper#(U52(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U52(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = proper#(X1) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X2 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X3 = proper#(X3) proper#(isNatIList(X)) = X >= X = proper#(X) proper#(U43(X)) = X >= X = proper#(X) proper#(U42(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U42(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X2 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X3 = proper#(X3) proper#(U32(X)) = X >= X = proper#(X) proper#(U31(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(isNat(X)) = X + 1 >= X = proper#(X) proper#(U22(X)) = X >= X = proper#(X) proper#(U21(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(isNatList(X)) = X >= X = proper#(X) proper#(U12(X)) = X >= X = proper#(X) proper#(U11(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 >= X2 = proper#(X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 + 1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 0 = mark(tt()) active(U31(tt(),V)) = V + 1 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 + 1 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 + 1 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 + 1 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 + 1 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 0 = mark(tt()) active(U61(tt(),L)) = L + 1 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= 0 = mark(X) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 2 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 + 2 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 + 1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 2 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 2 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 2 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 + 1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + 1 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X2 + 1 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X2 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X2 >= X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = X >= X = U12(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + X2 >= X1 + X2 = U42(proper(X1),proper(X2)) proper(U43(X)) = X >= X = U43(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + X2 >= X1 + X2 = U52(proper(X1),proper(X2)) proper(U53(X)) = X >= X = U53(proper(X)) proper(U61(X1,X2)) = X1 + X2 >= X1 + X2 = U61(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) isNat(ok(X)) = 1 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 0 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 0 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 0 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 0 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 0 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 1 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> proper#(X2) proper#(U53(X)) -> proper#(X) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(isNatIList(X)) -> proper#(X) proper#(U43(X)) -> proper#(X) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(isNatList(X)) -> proper#(X) proper#(U12(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) 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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 1, [and](x0, x1) = x0 + x1, [s](x0) = 1, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0 + 1, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: proper#(and(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(length(X)) = X >= X = proper#(X) proper#(U61(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U61(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U53(X)) = X >= X = proper#(X) proper#(U52(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U52(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = proper#(X1) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X2 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X3 = proper#(X3) proper#(isNatIList(X)) = X >= X = proper#(X) proper#(U43(X)) = X >= X = proper#(X) proper#(U42(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U42(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X2 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X3 = proper#(X3) proper#(U32(X)) = X >= X = proper#(X) proper#(U31(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U22(X)) = X >= X = proper#(X) proper#(U21(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(isNatList(X)) = X + 1 >= X = proper#(X) proper#(U12(X)) = X >= X = proper#(X) proper#(U11(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 >= X2 = proper#(X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 + 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = V + 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 + 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 + 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 + 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 + 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 + 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 2 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 2 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 2 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 + 2 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = 2 >= 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X2 + 1 >= 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 + 1 >= 1 = mark(U11(X1,X2)) U12(mark(X)) = 1 >= 1 = mark(U12(X)) U21(mark(X1),X2) = X2 + 1 >= 1 = mark(U21(X1,X2)) U22(mark(X)) = 1 >= 1 = mark(U22(X)) U31(mark(X1),X2) = X2 + 1 >= 1 = mark(U31(X1,X2)) U32(mark(X)) = 1 >= 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 + 1 >= 1 = mark(U42(X1,X2)) U43(mark(X)) = 1 >= 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X2 + X3 + 1 >= 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X2 + 1 >= 1 = mark(U52(X1,X2)) U53(mark(X)) = 1 >= 1 = mark(U53(X)) U61(mark(X1),X2) = X2 + 1 >= 1 = mark(U61(X1,X2)) s(mark(X)) = 1 >= 1 = mark(s(X)) length(mark(X)) = 1 >= 1 = mark(length(X)) and(mark(X1),X2) = X2 + 1 >= 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = X >= X = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + X2 >= X1 + X2 = U42(proper(X1),proper(X2)) proper(U43(X)) = X >= X = U43(proper(X)) proper(isNatIList(X)) = X >= X = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + X2 >= X1 + X2 = U52(proper(X1),proper(X2)) proper(U53(X)) = X >= X = U53(proper(X)) proper(U61(X1,X2)) = X1 + X2 >= X1 + X2 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(and(X1,X2)) = X1 + X2 >= X1 + X2 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 1 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 0 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 0 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 0 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 0 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 0 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 1 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(and(X1,X2)) -> proper#(X1) proper#(and(X1,X2)) -> proper#(X2) proper#(length(X)) -> proper#(X) proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> proper#(X2) proper#(U53(X)) -> proper#(X) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(isNatIList(X)) -> proper#(X) proper#(U43(X)) -> proper#(X) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(U12(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1 + 1, [s](x0) = x0, [length](x0) = x0 + 1, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0 + x1, [U51](x0, x1, x2) = x0 + x1 + x2, [U43](x0) = x0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x0 + x1 + x2, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: proper#(and(X1,X2)) = X1 + X2 + 1 >= X1 = proper#(X1) proper#(and(X1,X2)) = X1 + X2 + 1 >= X2 = proper#(X2) proper#(length(X)) = X + 1 >= X = proper#(X) proper#(U61(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U61(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U53(X)) = X >= X = proper#(X) proper#(U52(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U52(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 = proper#(X1) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X2 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 >= X3 = proper#(X3) proper#(isNatIList(X)) = X + 1 >= X = proper#(X) proper#(U43(X)) = X >= X = proper#(X) proper#(U42(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U42(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X2 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 >= X3 = proper#(X3) proper#(U32(X)) = X >= X = proper#(X) proper#(U31(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U22(X)) = X >= X = proper#(X) proper#(U21(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U12(X)) = X >= X = proper#(X) proper#(U11(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 >= X2 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = L >= 0 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 2 >= 0 = mark(0()) active(length(cons(N,L))) = 1 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + X2 >= X1 + X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + X2 >= X1 + X2 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 + X2 >= X1 + X2 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = X2 + X3 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X2 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = X2 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 1 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 + 1 >= 0 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = X >= X = U12(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + X2 >= X1 + X2 = U42(proper(X1),proper(X2)) proper(U43(X)) = X >= X = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + X2 >= X1 + X2 = U52(proper(X1),proper(X2)) proper(U53(X)) = X >= X = U53(proper(X)) proper(U61(X1,X2)) = X1 + X2 >= X1 + X2 = U61(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = X >= X = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = X >= X = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(U61(X1,X2)) -> proper#(X1) proper#(U61(X1,X2)) -> proper#(X2) proper#(U53(X)) -> proper#(X) proper#(U52(X1,X2)) -> proper#(X1) proper#(U52(X1,X2)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X1) proper#(U51(X1,X2,X3)) -> proper#(X2) proper#(U51(X1,X2,X3)) -> proper#(X3) proper#(U43(X)) -> proper#(X) proper#(U42(X1,X2)) -> proper#(X1) proper#(U42(X1,X2)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X1) proper#(U41(X1,X2,X3)) -> proper#(X2) proper#(U41(X1,X2,X3)) -> proper#(X3) proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(U12(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = 0, [length](x0) = 0, [U61](x0, x1) = x0 + x1 + 1, [U53](x0) = x0 + 1, [U52](x0, x1) = x0 + x1 + 1, [U51](x0, x1, x2) = x0 + x1 + x2 + 1, [U43](x0) = x0 + 1, [isNatIList](x0) = 0, [U42](x0, x1) = x0 + x1 + 1, [U41](x0, x1, x2) = x0 + x1 + x2 + 1, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: proper#(U61(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U61(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U53(X)) = X + 2 >= X + 1 = proper#(X) proper#(U52(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U52(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U51(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(U43(X)) = X + 2 >= X + 1 = proper#(X) proper#(U42(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U42(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U41(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(U32(X)) = X + 1 >= X + 1 = proper#(X) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U22(X)) = X + 1 >= X + 1 = proper#(X) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) proper#(U12(X)) = X + 1 >= X + 1 = proper#(X) proper#(U11(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = V1 + V2 + 1 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 + 1 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = V1 + V2 + 1 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = V2 + 1 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 0 = mark(tt()) active(U61(tt(),L)) = L + 1 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = X2 + X3 + 1 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 + 1 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 1 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = X2 + X3 + 1 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X2 + 1 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 1 >= 0 = mark(U53(X)) U61(mark(X1),X2) = X2 + 1 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = X >= X = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = X2 >= X2 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X >= X = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 0 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 1 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 1 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 1 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 1 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 1 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(U12(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [isNatKind](x0) = 1, [isNatIListKind](x0) = 0, [and](x0, x1) = 0, [s](x0) = x0, [length](x0) = 1, [U61](x0, x1) = 1, [U53](x0) = 1, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0 + x1, [U12](x0) = x0 + 1, [isNatList](x0) = x0 + 1, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 1, [active](x0) = x0, [zeros] = 1 orientation: proper#(U32(X)) = X >= X = proper#(X) proper#(U31(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U22(X)) = X >= X = proper#(X) proper#(U21(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U12(X)) = X + 1 >= X = proper#(X) proper#(U11(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 >= X2 = proper#(X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 0 = mark(tt()) active(U61(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = 1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 2 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = 1 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 1 >= 1 = U53(active(X)) active(U61(X1,X2)) = 1 >= 1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = 1 >= 1 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 1 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 1 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 1 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 1 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + X2 >= X1 + X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(isNat(X)) = X >= X = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = X >= X = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 0 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 1 >= 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 0 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 1 >= 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = 1 >= 1 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = 1 >= 1 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 1 >= 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U21(X1,X2)) -> proper#(X1) proper#(U21(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = 0, [s](x0) = 0, [length](x0) = 0, [U61](x0, x1) = 1, [U53](x0) = 0, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0, [isNat](x0) = x0 + 1, [U21](x0, x1) = x0 + x1 + 1, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 1, [active](x0) = x0, [zeros] = 1 orientation: proper#(U32(X)) = X >= X = proper#(X) proper#(U31(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U22(X)) = X >= X = proper#(X) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X1 = proper#(X1) proper#(U21(X1,X2)) = X1 + X2 + 1 >= X2 = proper#(X2) proper#(U11(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 >= X2 = proper#(X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 + 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(length(V1))) = 1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 1 >= 1 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 1 >= 1 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 + 1 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 1 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 1 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X >= X = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 0 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 1 >= 1 = ok(U52(X1,X2)) U53(ok(X)) = 0 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 1 >= 1 = ok(U61(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U22(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = 0, [s](x0) = 0, [length](x0) = x0, [U61](x0, x1) = 0, [U53](x0) = x0 + 1, [U52](x0, x1) = 0, [U51](x0, x1, x2) = x0 + 1, [U43](x0) = 1, [isNatIList](x0) = 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0 + x1, [U22](x0) = x0 + 1, [isNat](x0) = x0 + 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = x0 + x1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: proper#(U32(X)) = X >= X = proper#(X) proper#(U31(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 >= X2 = proper#(X2) proper#(U22(X)) = X + 1 >= X = proper#(X) proper#(U11(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 >= X2 = proper#(X2) active(zeros()) = 2 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 + 1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 2 >= 0 = mark(tt()) active(U31(tt(),V)) = V + 1 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 2 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 2 >= 0 = mark(tt()) active(U61(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 1 >= 0 = mark(X) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 2 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 2 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = 2 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 2 >= 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X)) = 1 >= 0 = U12(active(X)) active(U21(X1,X2)) = 1 >= 0 = U21(active(X1),X2) active(U22(X)) = X + 2 >= X + 2 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 1 >= 0 = U42(active(X1),X2) active(U43(X)) = 2 >= 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 1 >= 0 = U52(active(X1),X2) active(U53(X)) = X + 2 >= X + 2 = U53(active(X)) active(U61(X1,X2)) = 1 >= 0 = U61(active(X1),X2) active(s(X)) = 1 >= 0 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = 1 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 1 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 1 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 1 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 >= X1 + X2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 >= X1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = X >= X = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 1 >= 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 1 >= 1 = ok(U43(X)) isNatIList(ok(X)) = 1 >= 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 0 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = 0 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(U32(X)) -> proper#(X) proper#(U31(X1,X2)) -> proper#(X1) proper#(U31(X1,X2)) -> proper#(X2) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [nil] = 1, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0 + 1, [and](x0, x1) = x0 + 1, [s](x0) = 1, [length](x0) = 1, [U61](x0, x1) = 1, [U53](x0) = 0, [U52](x0, x1) = 1, [U51](x0, x1, x2) = 0, [U43](x0) = 1, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 1, [U41](x0, x1, x2) = 0, [U32](x0) = x0, [U31](x0, x1) = x0 + x1 + 1, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0 + x1, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: proper#(U32(X)) = X >= X = proper#(X) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X1 = proper#(X1) proper#(U31(X1,X2)) = X1 + X2 + 1 >= X2 = proper#(X2) proper#(U11(X1,X2)) = X1 + X2 >= X1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 >= X2 = proper#(X2) active(zeros()) = 1 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 + 1 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V + 2 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 1 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 2 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = 2 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 2 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 2 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 2 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 2 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = 1 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 1 >= 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 1 >= 1 = U42(active(X1),X2) active(U43(X)) = 1 >= 1 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 1 >= 1 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 1 >= 1 = U61(active(X1),X2) active(s(X)) = 1 >= 1 = s(active(X)) active(length(X)) = 1 >= 1 = length(active(X)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = X2 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 + 1 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 1 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 1 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 1 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 1 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) length(mark(X)) = 1 >= 0 = mark(length(X)) and(mark(X1),X2) = 1 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 >= X1 + X2 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 0 = ok(tt()) proper(U12(X)) = X >= X = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = X2 >= X2 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X >= X = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X >= X = isNatKind(proper(X)) proper(nil()) = 1 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 1 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) isNat(ok(X)) = 1 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 1 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 0 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 1 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 1 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 1 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 1 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 0 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 1 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) length(ok(X)) = 1 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 1 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 1 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(U32(X)) -> proper#(X) proper#(U11(X1,X2)) -> proper#(X1) proper#(U11(X1,X2)) -> proper#(X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = 1, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0 + 1, [U61](x0, x1) = x1 + 1, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = x0 + 1, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 1, [U32](x0) = x0 + 1, [U31](x0, x1) = x1 + 1, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 0, [U12](x0) = x0 + 1, [isNatList](x0) = x0, [U11](x0, x1) = x0 + x1 + 1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: proper#(U32(X)) = X + 2 >= X + 1 = proper#(X) proper#(U11(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U11(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = V1 + 1 >= V1 + 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = V + 1 >= V + 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X >= X = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 2 >= V1 + 2 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V + 1 >= V + 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 + 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 + 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= V1 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = 1 >= 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = X2 >= X2 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = X2 + 1 >= X2 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = 1 >= 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = X2 + 1 >= X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X >= X = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X2 >= X2 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X2 >= X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X >= X = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X2 + 1 >= X2 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = X2 + 1 >= X2 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X2 >= X2 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X >= X = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 1 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 1 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) isNat(ok(X)) = 1 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 1 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 1 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 0 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 1 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 0 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 0 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 1 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 1 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 1 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: active#(and(X1,X2)) -> active#(X1) active#(length(X)) -> active#(X) active#(s(X)) -> active#(X) active#(U61(X1,X2)) -> active#(X1) active#(U53(X)) -> active#(X) active#(U52(X1,X2)) -> active#(X1) active#(U51(X1,X2,X3)) -> active#(X1) active#(U43(X)) -> active#(X) active#(U42(X1,X2)) -> active#(X1) active#(U41(X1,X2,X3)) -> active#(X1) active#(U32(X)) -> active#(X) active#(U31(X1,X2)) -> active#(X1) active#(U22(X)) -> active#(X) active#(U21(X1,X2)) -> active#(X1) active#(U12(X)) -> active#(X) active#(U11(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> active#(X1) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0 + 1, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [U61](x0, x1) = x0 + 1, [U53](x0) = x0 + 1, [U52](x0, x1) = x0 + 1, [U51](x0, x1, x2) = x0 + 1, [U43](x0) = x0 + 1, [isNatIList](x0) = x0, [U42](x0, x1) = x0 + 1, [U41](x0, x1, x2) = x0 + 1, [U32](x0) = x0 + 1, [U31](x0, x1) = x0 + 1, [U22](x0) = x0 + 1, [isNat](x0) = x0, [U21](x0, x1) = x0 + 1, [U12](x0) = x0 + 1, [isNatList](x0) = 0, [U11](x0, x1) = x0 + 1, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: active#(and(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(length(X)) = X + 2 >= X + 1 = active#(X) active#(s(X)) = X + 2 >= X + 1 = active#(X) active#(U61(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U53(X)) = X + 2 >= X + 1 = active#(X) active#(U52(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U51(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U43(X)) = X + 2 >= X + 1 = active#(X) active#(U42(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U41(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U32(X)) = X + 2 >= X + 1 = active#(X) active#(U31(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U22(X)) = X + 2 >= X + 1 = active#(X) active#(U21(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U12(X)) = X + 2 >= X + 1 = active#(X) active#(U11(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active#(X1) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 2 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 2 >= 0 = mark(tt()) active(U21(tt(),V1)) = 2 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 2 >= 0 = mark(tt()) active(U31(tt(),V)) = 2 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 2 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 2 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 2 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 2 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 2 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 2 >= 0 = mark(tt()) active(U61(tt(),L)) = 2 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 2 >= 0 = mark(X) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 + 1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 + 1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 + 1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = N + 1 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 1 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 1 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 1 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 1 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 1 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 1 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 1 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 1 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 1 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 1 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 1 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 1 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 1 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) length(mark(X)) = 1 >= 0 = mark(length(X)) and(mark(X1),X2) = 1 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 0 = ok(0()) proper(U11(X1,X2)) = X1 + 2 >= X1 + 2 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 0 = ok(tt()) proper(U12(X)) = X + 2 >= X + 2 = U12(proper(X)) proper(isNatList(X)) = 1 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 2 >= X1 + 2 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 2 >= X + 2 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 2 >= X1 + 2 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 2 >= X + 2 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 2 >= X1 + 2 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 2 >= X + 2 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 2 >= X1 + 2 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 2 >= X1 + 2 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 2 >= X + 2 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 2 >= X1 + 2 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(length(X)) = X + 2 >= X + 2 = length(proper(X)) proper(and(X1,X2)) = X1 + 2 >= X1 + 2 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 1 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 1 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 1 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 1 >= 0 = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 1 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 1 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 1 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 1 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 1 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 1 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 1 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 1 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 1 >= 0 = ok(s(X)) length(ok(X)) = 1 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 1 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(cons(X1,X2)) -> active#(X1) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = 0, [s](x0) = 1, [length](x0) = 0, [U61](x0, x1) = 0, [U53](x0) = 0, [U52](x0, x1) = 0, [U51](x0, x1, x2) = 0, [U43](x0) = 0, [isNatIList](x0) = 0, [U42](x0, x1) = 0, [U41](x0, x1, x2) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U22](x0) = 0, [isNat](x0) = x0 + 1, [U21](x0, x1) = 0, [U12](x0) = 0, [isNatList](x0) = 0, [U11](x0, x1) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + 1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 2 >= 0 = mark(tt()) active(isNat(length(V1))) = 1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 2 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 1 >= 1 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 1 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 0 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 0 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 0 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 0 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 0 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 1 >= 1 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatIList#(ok(X)) -> isNatIList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatIList#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: isNatIList#(ok(X)) = X + 2 >= X + 1 = isNatIList#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatKind#(ok(X)) -> isNatKind#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatKind#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: isNatKind#(ok(X)) = X + 2 >= X + 1 = isNatKind#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNat#(ok(X)) -> isNat#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: isNat#(ok(X)) = X + 2 >= X + 1 = isNat#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatIListKind#(ok(X)) -> isNatIListKind#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatIListKind#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: isNatIListKind#(ok(X)) = X + 2 >= X + 1 = isNatIListKind#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: isNatList#(ok(X)) -> isNatList#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatList#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: isNatList#(ok(X)) = X + 2 >= X + 1 = isNatList#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(ok(X1),ok(X2)) -> cons#(X1,X2) cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: cons#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X1 + 2 >= X1 + 1 = cons#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: cons#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: cons#(ok(X1),ok(X2)) = X1 + 1 >= X1 = cons#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(ok(X1),ok(X2)) -> U11#(X1,X2) U11#(mark(X1),X2) -> U11#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U11#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = U11#(X1,X2) U11#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U11#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U11#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U11#(ok(X1),ok(X2)) = X1 + 1 >= X1 = U11#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U12#(ok(X)) -> U12#(X) U12#(mark(X)) -> U12#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U12#(ok(X)) = X + 1 >= X + 1 = U12#(X) U12#(mark(X)) = X + 2 >= X + 1 = U12#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U12#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U12#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: U12#(ok(X)) = X + 2 >= X + 1 = U12#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U21#(ok(X1),ok(X2)) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U21#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = U21#(X1,X2) U21#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U21#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U21#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U21#(ok(X1),ok(X2)) = X1 + 1 >= X1 = U21#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U22#(ok(X)) -> U22#(X) U22#(mark(X)) -> U22#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U22#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U22#(ok(X)) = X + 1 >= X + 1 = U22#(X) U22#(mark(X)) = X + 2 >= X + 1 = U22#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U22#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U22#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: U22#(ok(X)) = X + 2 >= X + 1 = U22#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U31#(ok(X1),ok(X2)) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U31#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = U31#(X1,X2) U31#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U31#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U31#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U31#(ok(X1),ok(X2)) = X1 + 1 >= X1 = U31#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U32#(ok(X)) -> U32#(X) U32#(mark(X)) -> U32#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U32#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U32#(ok(X)) = X + 1 >= X + 1 = U32#(X) U32#(mark(X)) = X + 2 >= X + 1 = U32#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U32#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U32#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: U32#(ok(X)) = X + 2 >= X + 1 = U32#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U41#(ok(X1),ok(X2),ok(X3)) -> U41#(X1,X2,X3) U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1, x2) = x1 + x2, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U41#(ok(X1),ok(X2),ok(X3)) = X2 + X3 + 2 >= X2 + X3 = U41#(X1,X2,X3) U41#(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U41#(X1,X2,X3) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1, x2) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U41#(mark(X1),X2,X3) = X1 + 2 >= X1 + 1 = U41#(X1,X2,X3) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 0 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 0 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 0 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 0 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 0 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U42#(ok(X1),ok(X2)) -> U42#(X1,X2) U42#(mark(X1),X2) -> U42#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U42#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U42#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = U42#(X1,X2) U42#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U42#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U42#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U42#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U42#(ok(X1),ok(X2)) = X1 + 1 >= X1 = U42#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U43#(ok(X)) -> U43#(X) U43#(mark(X)) -> U43#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U43#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U43#(ok(X)) = X + 1 >= X + 1 = U43#(X) U43#(mark(X)) = X + 2 >= X + 1 = U43#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U43#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U43#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: U43#(ok(X)) = X + 2 >= X + 1 = U43#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U51#(ok(X1),ok(X2),ok(X3)) -> U51#(X1,X2,X3) U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1, x2) = x1 + x2, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U51#(ok(X1),ok(X2),ok(X3)) = X2 + X3 + 2 >= X2 + X3 = U51#(X1,X2,X3) U51#(mark(X1),X2,X3) = X2 + X3 >= X2 + X3 = U51#(X1,X2,X3) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: U51#(mark(X1),X2,X3) -> U51#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1, x2) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U51#(mark(X1),X2,X3) = X1 + 2 >= X1 + 1 = U51#(X1,X2,X3) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = 0 >= 0 = ok(U11(X1,X2)) U12(ok(X)) = 0 >= 0 = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = 0 >= 0 = ok(U21(X1,X2)) U22(ok(X)) = 0 >= 0 = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = 0 >= 0 = ok(U31(X1,X2)) U32(ok(X)) = 0 >= 0 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = 0 >= 0 = ok(U42(X1,X2)) U43(ok(X)) = 0 >= 0 = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = 0 >= 0 = ok(U52(X1,X2)) U53(ok(X)) = 0 >= 0 = ok(U53(X)) U61(ok(X1),ok(X2)) = 0 >= 0 = ok(U61(X1,X2)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) and(ok(X1),ok(X2)) = 0 >= 0 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U52#(ok(X1),ok(X2)) -> U52#(X1,X2) U52#(mark(X1),X2) -> U52#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U52#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U52#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = U52#(X1,X2) U52#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U52#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U52#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U52#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U52#(ok(X1),ok(X2)) = X1 + 1 >= X1 = U52#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U53#(ok(X)) -> U53#(X) U53#(mark(X)) -> U53#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U53#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U53#(ok(X)) = X + 1 >= X + 1 = U53#(X) U53#(mark(X)) = X + 2 >= X + 1 = U53#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U53#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U53#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: U53#(ok(X)) = X + 2 >= X + 1 = U53#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U61#(ok(X1),ok(X2)) -> U61#(X1,X2) U61#(mark(X1),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U61#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = U61#(X1,X2) U61#(mark(X1),X2) = X1 + 2 >= X1 + 1 = U61#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: U61#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U61#(ok(X1),ok(X2)) = X1 + 1 >= X1 = U61#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(ok(X)) -> s#(X) s#(mark(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: s#(ok(X)) = X + 1 >= X + 1 = s#(X) s#(mark(X)) = X + 2 >= X + 1 = s#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: s#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length#(ok(X)) -> length#(X) length#(mark(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: length#(ok(X)) = X + 1 >= X + 1 = length#(X) length#(mark(X)) = X + 2 >= X + 1 = length#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: length#(ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x1, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x1, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: length#(ok(X)) = X + 2 >= X + 1 = length#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = V1 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = V2 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = X >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = V >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = V1 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = V1 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = V1 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = N >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 >= X1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 >= X1 = U11(active(X1),X2) active(U12(X)) = X >= X = U12(active(X)) active(U21(X1,X2)) = X2 >= X2 = U21(active(X1),X2) active(U22(X)) = X >= X = U22(active(X)) active(U31(X1,X2)) = X1 >= X1 = U31(active(X1),X2) active(U32(X)) = X >= X = U32(active(X)) active(U41(X1,X2,X3)) = X1 >= X1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X2 >= X2 = U42(active(X1),X2) active(U43(X)) = X >= X = U43(active(X)) active(U51(X1,X2,X3)) = X1 >= X1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 >= X1 = U52(active(X1),X2) active(U53(X)) = X >= X = U53(active(X)) active(U61(X1,X2)) = X1 >= X1 = U61(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) active(and(X1,X2)) = X2 >= X2 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = X2 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X2 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = X2 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = 1 >= 1 = U11(proper(X1),proper(X2)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U12(X)) = 1 >= 1 = U12(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U21(X1,X2)) = 1 >= 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = 1 >= 1 = U22(proper(X)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(U31(X1,X2)) = 1 >= 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = 1 >= 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = 1 >= 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 1 >= 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = 1 >= 1 = U43(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 1 >= 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 1 >= 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = 1 >= 1 = U53(proper(X)) proper(U61(X1,X2)) = 1 >= 1 = U61(proper(X1),proper(X2)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(and(X1,X2)) = 1 >= 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 1 >= 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 1 >= 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: and#(ok(X1),ok(X2)) -> and#(X1,X2) and#(mark(X1),X2) -> and#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = 0, [nil] = 0, [isNatKind](x0) = 0, [isNatIListKind](x0) = 0, [and](x0, x1) = x0 + x1, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0 + x1, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = 0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = 0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = 0, [U11](x0, x1) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: and#(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = and#(X1,X2) and#(mark(X1),X2) = X1 + 2 >= X1 + 1 = and#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 1 >= 1 = mark(U12(isNatList(V1))) active(U12(tt())) = 1 >= 1 = mark(tt()) active(U21(tt(),V1)) = 1 >= 1 = mark(U22(isNat(V1))) active(U22(tt())) = 1 >= 1 = mark(tt()) active(U31(tt(),V)) = 1 >= 1 = mark(U32(isNatList(V))) active(U32(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V1,V2)) = 1 >= 1 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 1 >= 1 = mark(U43(isNatIList(V2))) active(U43(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V1,V2)) = 1 >= 1 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 1 >= 1 = mark(U53(isNatList(V2))) active(U53(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(and(tt(),X)) = X + 1 >= X + 1 = mark(X) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 1 >= 1 = mark(tt()) active(isNatIListKind(zeros())) = 1 >= 1 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 1 >= 1 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 1 >= 1 = mark(tt()) active(isNatKind(length(V1))) = 1 >= 1 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 1 >= 1 = mark(isNatKind(V1)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(active(X1),X2) active(U12(X)) = X + 1 >= X + 1 = U12(active(X)) active(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(active(X1),X2) active(U22(X)) = X + 1 >= X + 1 = U22(active(X)) active(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(active(X1),X2) active(U32(X)) = X + 1 >= X + 1 = U32(active(X)) active(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2,X3) active(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(active(X1),X2) active(U43(X)) = X + 1 >= X + 1 = U43(active(X)) active(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2,X3) active(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(active(X1),X2) active(U53(X)) = X + 1 >= X + 1 = U53(active(X)) active(U61(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) active(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = and(active(X1),X2) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U11(X1,X2)) U12(mark(X)) = X + 1 >= X + 1 = mark(U12(X)) U21(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U21(X1,X2)) U22(mark(X)) = X + 1 >= X + 1 = mark(U22(X)) U31(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U31(X1,X2)) U32(mark(X)) = X + 1 >= X + 1 = mark(U32(X)) U41(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U42(X1,X2)) U43(mark(X)) = X + 1 >= X + 1 = mark(U43(X)) U51(mark(X1),X2,X3) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U52(X1,X2)) U53(mark(X)) = X + 1 >= X + 1 = mark(U53(X)) U61(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) and(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(and(X1,X2)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X1,X2)) = 0 >= 0 = U11(proper(X1),proper(X2)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U12(X)) = 0 >= 0 = U12(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U21(X1,X2)) = 0 >= 0 = U21(proper(X1),proper(X2)) proper(U22(X)) = 0 >= 0 = U22(proper(X)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(U31(X1,X2)) = 0 >= 0 = U31(proper(X1),proper(X2)) proper(U32(X)) = 0 >= 0 = U32(proper(X)) proper(U41(X1,X2,X3)) = 0 >= 0 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = 0 >= 0 = U42(proper(X1),proper(X2)) proper(U43(X)) = 0 >= 0 = U43(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = 0 >= 0 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = 0 >= 0 = U52(proper(X1),proper(X2)) proper(U53(X)) = 0 >= 0 = U53(proper(X)) proper(U61(X1,X2)) = 0 >= 0 = U61(proper(X1),proper(X2)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(and(X1,X2)) = 0 >= 0 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = 0 >= 0 = isNatIListKind(proper(X)) proper(isNatKind(X)) = 0 >= 0 = isNatKind(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 >= X1 = ok(U11(X1,X2)) U12(ok(X)) = X >= X = ok(U12(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 >= X1 = ok(U21(X1,X2)) U22(ok(X)) = X >= X = ok(U22(X)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 >= X1 = ok(U31(X1,X2)) U32(ok(X)) = X >= X = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 >= X1 = ok(U42(X1,X2)) U43(ok(X)) = X >= X = ok(U43(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 >= X1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 >= X1 = ok(U52(X1,X2)) U53(ok(X)) = X >= X = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(U61(X1,X2)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X >= X = ok(length(X)) and(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(and(X1,X2)) isNatIListKind(ok(X)) = 0 >= 0 = ok(isNatIListKind(X)) isNatKind(ok(X)) = 0 >= 0 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: and#(ok(X1),ok(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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dimension: 1 interpretation: [and#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [isNatKind](x0) = x0, [isNatIListKind](x0) = x0, [and](x0, x1) = x0, [s](x0) = x0, [length](x0) = x0, [U61](x0, x1) = x0, [U53](x0) = x0, [U52](x0, x1) = x0, [U51](x0, x1, x2) = x0, [U43](x0) = x0, [isNatIList](x0) = x0, [U42](x0, x1) = x0, [U41](x0, x1, x2) = x0, [U32](x0) = x0, [U31](x0, x1) = x0, [U22](x0) = x0, [isNat](x0) = x0, [U21](x0, x1) = x0, [U12](x0) = x0, [isNatList](x0) = x0, [U11](x0, x1) = x0, [tt] = 1, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: and#(ok(X1),ok(X2)) = X1 + 1 >= X1 = and#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt(),V1)) = 0 >= 0 = mark(U12(isNatList(V1))) active(U12(tt())) = 0 >= 0 = mark(tt()) active(U21(tt(),V1)) = 0 >= 0 = mark(U22(isNat(V1))) active(U22(tt())) = 0 >= 0 = mark(tt()) active(U31(tt(),V)) = 0 >= 0 = mark(U32(isNatList(V))) active(U32(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V1,V2)) = 0 >= 0 = mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) = 0 >= 0 = mark(U43(isNatIList(V2))) active(U43(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V1,V2)) = 0 >= 0 = mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) = 0 >= 0 = mark(U53(isNatList(V2))) active(U53(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(and(tt(),X)) = 0 >= 0 = mark(X) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatIListKind(V1),V1)) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNatKind(V1),V1)) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatIListKind(V),V)) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(isNatIListKind(nil())) = 0 >= 0 = mark(tt()) active(isNatIListKind(zeros())) = 0 >= 0 = mark(tt()) active(isNatIListKind(cons(V1,V2))) = 0 >= 0 = mark(and(isNatKind(V1),isNatIListKind(V2))) active(isNatKind(0())) = 0 >= 0 = mark(tt()) active(isNatKind(length(V1))) = 0 >= 0 = mark(isNatIListKind(V1)) active(isNatKind(s(V1))) = 0 >= 0 = mark(isNatKind(V1)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(and(isNatKind(V1),isNatIListKind(V2)),V1,V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(and(and(isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))), L)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X1,X2)) = 0 >= 0 = U11(active(X1),X2) active(U12(X)) = 0 >= 0 = U12(active(X)) active(U21(X1,X2)) = 0 >= 0 = U21(active(X1),X2) active(U22(X)) = 0 >= 0 = U22(active(X)) active(U31(X1,X2)) = 0 >= 0 = U31(active(X1),X2) active(U32(X)) = 0 >= 0 = U32(active(X)) active(U41(X1,X2,X3)) = 0 >= 0 = U41(active(X1),X2,X3) active(U42(X1,X2)) = 0 >= 0 = U42(active(X1),X2) active(U43(X)) = 0 >= 0 = U43(active(X)) active(U51(X1,X2,X3)) = 0 >= 0 = U51(active(X1),X2,X3) active(U52(X1,X2)) = 0 >= 0 = U52(active(X1),X2) active(U53(X)) = 0 >= 0 = U53(active(X)) active(U61(X1,X2)) = 0 >= 0 = U61(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) active(and(X1,X2)) = 0 >= 0 = and(active(X1),X2) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X1),X2) = 0 >= 0 = mark(U11(X1,X2)) U12(mark(X)) = 0 >= 0 = mark(U12(X)) U21(mark(X1),X2) = 0 >= 0 = mark(U21(X1,X2)) U22(mark(X)) = 0 >= 0 = mark(U22(X)) U31(mark(X1),X2) = 0 >= 0 = mark(U31(X1,X2)) U32(mark(X)) = 0 >= 0 = mark(U32(X)) U41(mark(X1),X2,X3) = 0 >= 0 = mark(U41(X1,X2,X3)) U42(mark(X1),X2) = 0 >= 0 = mark(U42(X1,X2)) U43(mark(X)) = 0 >= 0 = mark(U43(X)) U51(mark(X1),X2,X3) = 0 >= 0 = mark(U51(X1,X2,X3)) U52(mark(X1),X2) = 0 >= 0 = mark(U52(X1,X2)) U53(mark(X)) = 0 >= 0 = mark(U53(X)) U61(mark(X1),X2) = 0 >= 0 = mark(U61(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) and(mark(X1),X2) = 0 >= 0 = mark(and(X1,X2)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X1,X2)) = X1 + 1 >= X1 + 1 = U11(proper(X1),proper(X2)) proper(tt()) = 2 >= 2 = ok(tt()) proper(U12(X)) = X + 1 >= X + 1 = U12(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U21(X1,X2)) = X1 + 1 >= X1 + 1 = U21(proper(X1),proper(X2)) proper(U22(X)) = X + 1 >= X + 1 = U22(proper(X)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(U31(X1,X2)) = X1 + 1 >= X1 + 1 = U31(proper(X1),proper(X2)) proper(U32(X)) = X + 1 >= X + 1 = U32(proper(X)) proper(U41(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) = X1 + 1 >= X1 + 1 = U42(proper(X1),proper(X2)) proper(U43(X)) = X + 1 >= X + 1 = U43(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) = X1 + 1 >= X1 + 1 = U52(proper(X1),proper(X2)) proper(U53(X)) = X + 1 >= X + 1 = U53(proper(X)) proper(U61(X1,X2)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(and(X1,X2)) = X1 + 1 >= X1 + 1 = and(proper(X1),proper(X2)) proper(isNatIListKind(X)) = X + 1 >= X + 1 = isNatIListKind(proper(X)) proper(isNatKind(X)) = X + 1 >= X + 1 = isNatKind(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U11(X1,X2)) U12(ok(X)) = X + 1 >= X + 1 = ok(U12(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U21(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U21(X1,X2)) U22(ok(X)) = X + 1 >= X + 1 = ok(U22(X)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) U31(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U31(X1,X2)) U32(ok(X)) = X + 1 >= X + 1 = ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U42(X1,X2)) U43(ok(X)) = X + 1 >= X + 1 = ok(U43(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U52(X1,X2)) U53(ok(X)) = X + 1 >= X + 1 = ok(U53(X)) U61(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) and(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(and(X1,X2)) isNatIListKind(ok(X)) = X + 1 >= X + 1 = ok(isNatIListKind(X)) isNatKind(ok(X)) = X + 1 >= X + 1 = ok(isNatKind(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt(),V1)) -> mark(U12(isNatList(V1))) active(U12(tt())) -> mark(tt()) active(U21(tt(),V1)) -> mark(U22(isNat(V1))) active(U22(tt())) -> mark(tt()) active(U31(tt(),V)) -> mark(U32(isNatList(V))) active(U32(tt())) -> mark(tt()) active(U41(tt(),V1,V2)) -> mark(U42(isNat(V1),V2)) active(U42(tt(),V2)) -> mark(U43(isNatIList(V2))) active(U43(tt())) -> mark(tt()) active(U51(tt(),V1,V2)) -> mark(U52(isNat(V1),V2)) active(U52(tt(),V2)) -> mark(U53(isNatList(V2))) active(U53(tt())) -> mark(tt()) active(U61(tt(),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)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X1,X2)) -> U11(active(X1),X2) active(U12(X)) -> U12(active(X)) active(U21(X1,X2)) -> U21(active(X1),X2) active(U22(X)) -> U22(active(X)) active(U31(X1,X2)) -> U31(active(X1),X2) active(U32(X)) -> U32(active(X)) active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3) active(U42(X1,X2)) -> U42(active(X1),X2) active(U43(X)) -> U43(active(X)) active(U51(X1,X2,X3)) -> U51(active(X1),X2,X3) active(U52(X1,X2)) -> U52(active(X1),X2) active(U53(X)) -> U53(active(X)) active(U61(X1,X2)) -> U61(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(and(X1,X2)) -> and(active(X1),X2) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X1),X2) -> mark(U11(X1,X2)) U12(mark(X)) -> mark(U12(X)) U21(mark(X1),X2) -> mark(U21(X1,X2)) U22(mark(X)) -> mark(U22(X)) U31(mark(X1),X2) -> mark(U31(X1,X2)) U32(mark(X)) -> mark(U32(X)) U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3)) U42(mark(X1),X2) -> mark(U42(X1,X2)) U43(mark(X)) -> mark(U43(X)) U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3)) U52(mark(X1),X2) -> mark(U52(X1,X2)) U53(mark(X)) -> mark(U53(X)) U61(mark(X1),X2) -> mark(U61(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X1,X2)) -> U11(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(U12(X)) -> U12(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) proper(U22(X)) -> U22(proper(X)) proper(isNat(X)) -> isNat(proper(X)) proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) proper(U32(X)) -> U32(proper(X)) proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3)) proper(U42(X1,X2)) -> U42(proper(X1),proper(X2)) proper(U43(X)) -> U43(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2,X3)) -> U51(proper(X1),proper(X2),proper(X3)) proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) proper(U53(X)) -> U53(proper(X)) proper(U61(X1,X2)) -> U61(proper(X1),proper(X2)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) proper(isNatKind(X)) -> isNatKind(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X1),ok(X2)) -> ok(U11(X1,X2)) U12(ok(X)) -> ok(U12(X)) isNatList(ok(X)) -> ok(isNatList(X)) U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) U22(ok(X)) -> ok(U22(X)) isNat(ok(X)) -> ok(isNat(X)) U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) U32(ok(X)) -> ok(U32(X)) U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3)) U42(ok(X1),ok(X2)) -> ok(U42(X1,X2)) U43(ok(X)) -> ok(U43(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3)) U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) U53(ok(X)) -> ok(U53(X)) U61(ok(X1),ok(X2)) -> ok(U61(X1,X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) isNatKind(ok(X)) -> ok(isNatKind(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed