MAYBE Problem: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U41(tt(),V2)) -> isNatIList#(V2) active#(U41(tt(),V2)) -> U42#(isNatIList(V2)) active#(U51(tt(),V2)) -> isNatList#(V2) active#(U51(tt(),V2)) -> U52#(isNatList(V2)) active#(U61(tt(),L,N)) -> isNat#(N) active#(U61(tt(),L,N)) -> U62#(isNat(N),L) active#(U62(tt(),L)) -> length#(L) active#(U62(tt(),L)) -> s#(length(L)) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(length(V1))) -> U11#(isNatList(V1)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> U21#(isNat(V1)) active#(isNatIList(V)) -> isNatList#(V) active#(isNatIList(V)) -> U31#(isNatList(V)) active#(isNatIList(cons(V1,V2))) -> isNat#(V1) active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2) active#(isNatList(cons(V1,V2))) -> isNat#(V1) active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> U61#(isNatList(L),L,N) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X)) -> active#(X) active#(U11(X)) -> U11#(active(X)) active#(U21(X)) -> active#(X) active#(U21(X)) -> U21#(active(X)) active#(U31(X)) -> active#(X) active#(U31(X)) -> U31#(active(X)) active#(U41(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> U41#(active(X1),X2) active#(U42(X)) -> active#(X) active#(U42(X)) -> U42#(active(X)) active#(U51(X1,X2)) -> active#(X1) active#(U51(X1,X2)) -> U51#(active(X1),X2) active#(U52(X)) -> active#(X) active#(U52(X)) -> U52#(active(X)) active#(U61(X1,X2,X3)) -> active#(X1) active#(U61(X1,X2,X3)) -> U61#(active(X1),X2,X3) active#(U62(X1,X2)) -> active#(X1) active#(U62(X1,X2)) -> U62#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X)) -> U11#(X) U21#(mark(X)) -> U21#(X) U31#(mark(X)) -> U31#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U42#(mark(X)) -> U42#(X) U51#(mark(X1),X2) -> U51#(X1,X2) U52#(mark(X)) -> U52#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X)) -> proper#(X) proper#(U11(X)) -> U11#(proper(X)) proper#(U21(X)) -> proper#(X) proper#(U21(X)) -> U21#(proper(X)) proper#(U31(X)) -> proper#(X) proper#(U31(X)) -> U31#(proper(X)) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2)) proper#(U42(X)) -> proper#(X) proper#(U42(X)) -> U42#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2)) -> proper#(X2) proper#(U51(X1,X2)) -> proper#(X1) proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2)) proper#(U52(X)) -> proper#(X) proper#(U52(X)) -> U52#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> U61#(proper(X1),proper(X2),proper(X3)) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> U62#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X)) -> U11#(X) U21#(ok(X)) -> U21#(X) U31#(ok(X)) -> U31#(X) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) U42#(ok(X)) -> U42#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2)) -> U51#(X1,X2) U52#(ok(X)) -> U52#(X) isNatList#(ok(X)) -> isNatList#(X) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) isNat#(ok(X)) -> isNat#(X) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(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())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) TDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) active#(U41(tt(),V2)) -> isNatIList#(V2) active#(U41(tt(),V2)) -> U42#(isNatIList(V2)) active#(U51(tt(),V2)) -> isNatList#(V2) active#(U51(tt(),V2)) -> U52#(isNatList(V2)) active#(U61(tt(),L,N)) -> isNat#(N) active#(U61(tt(),L,N)) -> U62#(isNat(N),L) active#(U62(tt(),L)) -> length#(L) active#(U62(tt(),L)) -> s#(length(L)) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(length(V1))) -> U11#(isNatList(V1)) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> U21#(isNat(V1)) active#(isNatIList(V)) -> isNatList#(V) active#(isNatIList(V)) -> U31#(isNatList(V)) active#(isNatIList(cons(V1,V2))) -> isNat#(V1) active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2) active#(isNatList(cons(V1,V2))) -> isNat#(V1) active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> U61#(isNatList(L),L,N) active#(cons(X1,X2)) -> active#(X1) active#(cons(X1,X2)) -> cons#(active(X1),X2) active#(U11(X)) -> active#(X) active#(U11(X)) -> U11#(active(X)) active#(U21(X)) -> active#(X) active#(U21(X)) -> U21#(active(X)) active#(U31(X)) -> active#(X) active#(U31(X)) -> U31#(active(X)) active#(U41(X1,X2)) -> active#(X1) active#(U41(X1,X2)) -> U41#(active(X1),X2) active#(U42(X)) -> active#(X) active#(U42(X)) -> U42#(active(X)) active#(U51(X1,X2)) -> active#(X1) active#(U51(X1,X2)) -> U51#(active(X1),X2) active#(U52(X)) -> active#(X) active#(U52(X)) -> U52#(active(X)) active#(U61(X1,X2,X3)) -> active#(X1) active#(U61(X1,X2,X3)) -> U61#(active(X1),X2,X3) active#(U62(X1,X2)) -> active#(X1) active#(U62(X1,X2)) -> U62#(active(X1),X2) active#(s(X)) -> active#(X) active#(s(X)) -> s#(active(X)) active#(length(X)) -> active#(X) active#(length(X)) -> length#(active(X)) cons#(mark(X1),X2) -> cons#(X1,X2) U11#(mark(X)) -> U11#(X) U21#(mark(X)) -> U21#(X) U31#(mark(X)) -> U31#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U42#(mark(X)) -> U42#(X) U51#(mark(X1),X2) -> U51#(X1,X2) U52#(mark(X)) -> U52#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) s#(mark(X)) -> s#(X) length#(mark(X)) -> length#(X) proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) proper#(U11(X)) -> proper#(X) proper#(U11(X)) -> U11#(proper(X)) proper#(U21(X)) -> proper#(X) proper#(U21(X)) -> U21#(proper(X)) proper#(U31(X)) -> proper#(X) proper#(U31(X)) -> U31#(proper(X)) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> U41#(proper(X1),proper(X2)) proper#(U42(X)) -> proper#(X) proper#(U42(X)) -> U42#(proper(X)) proper#(isNatIList(X)) -> proper#(X) proper#(isNatIList(X)) -> isNatIList#(proper(X)) proper#(U51(X1,X2)) -> proper#(X2) proper#(U51(X1,X2)) -> proper#(X1) proper#(U51(X1,X2)) -> U51#(proper(X1),proper(X2)) proper#(U52(X)) -> proper#(X) proper#(U52(X)) -> U52#(proper(X)) proper#(isNatList(X)) -> proper#(X) proper#(isNatList(X)) -> isNatList#(proper(X)) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> U61#(proper(X1),proper(X2),proper(X3)) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> U62#(proper(X1),proper(X2)) proper#(isNat(X)) -> proper#(X) proper#(isNat(X)) -> isNat#(proper(X)) proper#(s(X)) -> proper#(X) proper#(s(X)) -> s#(proper(X)) proper#(length(X)) -> proper#(X) proper#(length(X)) -> length#(proper(X)) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) U11#(ok(X)) -> U11#(X) U21#(ok(X)) -> U21#(X) U31#(ok(X)) -> U31#(X) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) U42#(ok(X)) -> U42#(X) isNatIList#(ok(X)) -> isNatIList#(X) U51#(ok(X1),ok(X2)) -> U51#(X1,X2) U52#(ok(X)) -> U52#(X) isNatList#(ok(X)) -> isNatList#(X) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) isNat#(ok(X)) -> isNat#(X) s#(ok(X)) -> s#(X) length#(ok(X)) -> length#(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())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) graph: ... SCC Processor: #sccs: 18 #rules: 62 #arcs: 1520/12544 DPs: top#(ok(X)) -> top#(active(X)) top#(mark(X)) -> top#(proper(X)) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Open DPs: active#(cons(X1,X2)) -> active#(X1) active#(U11(X)) -> active#(X) active#(U21(X)) -> active#(X) active#(U31(X)) -> active#(X) active#(U41(X1,X2)) -> active#(X1) active#(U42(X)) -> active#(X) active#(U51(X1,X2)) -> active#(X1) active#(U52(X)) -> active#(X) active#(U61(X1,X2,X3)) -> active#(X1) active#(U62(X1,X2)) -> active#(X1) active#(s(X)) -> active#(X) active#(length(X)) -> active#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(active#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: proper#(cons(X1,X2)) -> proper#(X2) proper#(cons(X1,X2)) -> proper#(X1) proper#(U11(X)) -> proper#(X) proper#(U21(X)) -> proper#(X) proper#(U31(X)) -> proper#(X) proper#(U41(X1,X2)) -> proper#(X2) proper#(U41(X1,X2)) -> proper#(X1) proper#(U42(X)) -> proper#(X) proper#(isNatIList(X)) -> proper#(X) proper#(U51(X1,X2)) -> proper#(X2) proper#(U51(X1,X2)) -> proper#(X1) proper#(U52(X)) -> proper#(X) proper#(isNatList(X)) -> proper#(X) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U62(X1,X2)) -> proper#(X2) proper#(U62(X1,X2)) -> proper#(X1) proper#(isNat(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(length(X)) -> proper#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(proper#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: length#(mark(X)) -> length#(X) length#(ok(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(length#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: s#(mark(X)) -> s#(X) s#(ok(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(s#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(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())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNat#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U62#(mark(X1),X2) -> U62#(X1,X2) U62#(ok(X1),ok(X2)) -> U62#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U62#) = 1 problem: DPs: U62#(mark(X1),X2) -> U62#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U62#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U61#) = 2 problem: DPs: U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U61#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(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())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNatList#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U52#(mark(X)) -> U52#(X) U52#(ok(X)) -> U52#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U52#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U51#(mark(X1),X2) -> U51#(X1,X2) U51#(ok(X1),ok(X2)) -> U51#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U51#) = 1 problem: DPs: U51#(mark(X1),X2) -> U51#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U51#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(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())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(isNatIList#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U42#(mark(X)) -> U42#(X) U42#(ok(X)) -> U42#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U42#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U41#(mark(X1),X2) -> U41#(X1,X2) U41#(ok(X1),ok(X2)) -> U41#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U41#) = 1 problem: DPs: U41#(mark(X1),X2) -> U41#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U41#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U31#(mark(X)) -> U31#(X) U31#(ok(X)) -> U31#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U31#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U21#(mark(X)) -> U21#(X) U21#(ok(X)) -> U21#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U21#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: U11#(mark(X)) -> U11#(X) U11#(ok(X)) -> U11#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(U11#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed DPs: cons#(mark(X1),X2) -> cons#(X1,X2) cons#(ok(X1),ok(X2)) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 1 problem: DPs: cons#(mark(X1),X2) -> cons#(X1,X2) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Subterm Criterion Processor: simple projection: pi(cons#) = 0 problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> mark(tt()) active(U21(tt())) -> mark(tt()) active(U31(tt())) -> mark(tt()) active(U41(tt(),V2)) -> mark(U42(isNatIList(V2))) active(U42(tt())) -> mark(tt()) active(U51(tt(),V2)) -> mark(U52(isNatList(V2))) active(U52(tt())) -> mark(tt()) active(U61(tt(),L,N)) -> mark(U62(isNat(N),L)) active(U62(tt(),L)) -> mark(s(length(L))) active(isNat(0())) -> mark(tt()) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros())) -> mark(tt()) active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2)) active(isNatList(nil())) -> mark(tt()) active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2)) active(length(nil())) -> mark(0()) active(length(cons(N,L))) -> mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) -> cons(active(X1),X2) active(U11(X)) -> U11(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X)) -> U31(active(X)) active(U41(X1,X2)) -> U41(active(X1),X2) active(U42(X)) -> U42(active(X)) active(U51(X1,X2)) -> U51(active(X1),X2) active(U52(X)) -> U52(active(X)) active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) active(U62(X1,X2)) -> U62(active(X1),X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) U11(mark(X)) -> mark(U11(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X)) -> mark(U31(X)) U41(mark(X1),X2) -> mark(U41(X1,X2)) U42(mark(X)) -> mark(U42(X)) U51(mark(X1),X2) -> mark(U51(X1,X2)) U52(mark(X)) -> mark(U52(X)) U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) U62(mark(X1),X2) -> mark(U62(X1,X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) proper(zeros()) -> ok(zeros()) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(0()) -> ok(0()) proper(U11(X)) -> U11(proper(X)) proper(tt()) -> ok(tt()) proper(U21(X)) -> U21(proper(X)) proper(U31(X)) -> U31(proper(X)) proper(U41(X1,X2)) -> U41(proper(X1),proper(X2)) proper(U42(X)) -> U42(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) proper(U52(X)) -> U52(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) -> U62(proper(X1),proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(nil()) -> ok(nil()) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) U11(ok(X)) -> ok(U11(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X)) -> ok(U31(X)) U41(ok(X1),ok(X2)) -> ok(U41(X1,X2)) U42(ok(X)) -> ok(U42(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) U52(ok(X)) -> ok(U52(X)) isNatList(ok(X)) -> ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) -> ok(U62(X1,X2)) isNat(ok(X)) -> ok(isNat(X)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Qed