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)) EDG 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: 1514/12544 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: proper#(length(X)) -> proper#(X) proper#(s(X)) -> proper#(X) proper#(isNat(X)) -> proper#(X) proper#(U62(X1,X2)) -> proper#(X1) proper#(U62(X1,X2)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X1) proper#(U61(X1,X2,X3)) -> proper#(X2) proper#(U61(X1,X2,X3)) -> proper#(X3) proper#(isNatList(X)) -> proper#(X) proper#(U52(X)) -> proper#(X) proper#(U51(X1,X2)) -> proper#(X1) proper#(U51(X1,X2)) -> proper#(X2) proper#(isNatIList(X)) -> proper#(X) proper#(U42(X)) -> proper#(X) proper#(U41(X1,X2)) -> proper#(X1) proper#(U41(X1,X2)) -> proper#(X2) proper#(U31(X)) -> proper#(X) proper#(U21(X)) -> proper#(X) proper#(U11(X)) -> proper#(X) proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = x0 + 1, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + x1 + 1, [isNat](x0) = x0 + 1, [U61](x0, x1, x2) = x0 + x1 + x2 + 1, [U52](x0) = x0 + 1, [isNatList](x0) = x0 + 1, [U51](x0, x1) = x0 + x1 + 1, [U42](x0) = x0 + 1, [isNatIList](x0) = x0 + 1, [U41](x0, x1) = x0 + x1 + 1, [U31](x0) = x0 + 1, [U21](x0) = x0 + 1, [U11](x0) = x0 + 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + x1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: proper#(length(X)) = X + 2 >= X + 1 = proper#(X) proper#(s(X)) = X + 2 >= X + 1 = proper#(X) proper#(isNat(X)) = X + 2 >= X + 1 = proper#(X) proper#(U62(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U62(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U61(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = proper#(X1) proper#(U61(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X2 + 1 = proper#(X2) proper#(U61(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X3 + 1 = proper#(X3) proper#(isNatList(X)) = X + 2 >= X + 1 = proper#(X) proper#(U52(X)) = X + 2 >= X + 1 = proper#(X) proper#(U51(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U51(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(isNatIList(X)) = X + 2 >= X + 1 = proper#(X) proper#(U42(X)) = X + 2 >= X + 1 = proper#(X) proper#(U41(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(U41(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) proper#(U31(X)) = X + 2 >= X + 1 = proper#(X) proper#(U21(X)) = X + 2 >= X + 1 = proper#(X) proper#(U11(X)) = X + 2 >= X + 1 = proper#(X) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = proper#(X2) active(zeros()) = 2 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 2 >= 0 = mark(tt()) active(U21(tt())) = 2 >= 0 = mark(tt()) active(U31(tt())) = 2 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 2 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 2 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 + 2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 2 >= 0 = mark(tt()) active(U61(tt(),L,N)) = L + N + 2 >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 2 >= 0 = mark(s(length(L))) active(isNat(0())) = 3 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 3 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 + 3 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V + 2 >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 3 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + V2 + 2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 2 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 + 2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 2 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 2 >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 2 >= X + 2 = U11(active(X)) active(U21(X)) = X + 2 >= X + 2 = U21(active(X)) active(U31(X)) = X + 2 >= X + 2 = U31(active(X)) active(U41(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U41(active(X1),X2) active(U42(X)) = X + 2 >= X + 2 = U42(active(X)) active(U51(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U51(active(X1),X2) active(U52(X)) = X + 2 >= X + 2 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U62(active(X1),X2) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 1 >= 0 = mark(U11(X)) U21(mark(X)) = 1 >= 0 = mark(U21(X)) U31(mark(X)) = 1 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 1 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 + 1 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 1 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 + 1 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 1 >= 0 = mark(s(X)) length(mark(X)) = 1 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: proper#(cons(X1,X2)) -> proper#(X1) proper#(cons(X1,X2)) -> proper#(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [proper#](x0) = x0 + 1, [top](x0) = 1, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = 0, [length](x0) = 0, [U62](x0, x1) = 0, [isNat](x0) = 0, [U61](x0, x1, x2) = 0, [U52](x0) = 0, [isNatList](x0) = 0, [U51](x0, x1) = 0, [U42](x0) = 0, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = 0, [U11](x0) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0 + x1 + 1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: proper#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = proper#(X1) proper#(cons(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = proper#(X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = 0 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 0 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = 0 >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = 0 >= 0 = U11(active(X)) active(U21(X)) = 0 >= 0 = U21(active(X)) active(U31(X)) = 0 >= 0 = U31(active(X)) active(U41(X1,X2)) = 0 >= 0 = U41(active(X1),X2) active(U42(X)) = 0 >= 0 = U42(active(X)) active(U51(X1,X2)) = 0 >= 0 = U51(active(X1),X2) active(U52(X)) = 0 >= 0 = U52(active(X)) active(U61(X1,X2,X3)) = 0 >= 0 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 0 >= 0 = U62(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) cons(mark(X1),X2) = X2 + 1 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = 0 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = 0 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 0 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 0 >= 0 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(U31(X)) = 0 >= 0 = U31(proper(X)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(U42(X)) = 0 >= 0 = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = 0 >= 0 = U51(proper(X1),proper(X2)) proper(U52(X)) = 0 >= 0 = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 0 >= 0 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 0 >= 0 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = 0 >= 0 = ok(U11(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X)) = 0 >= 0 = ok(U31(X)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) U42(ok(X)) = 0 >= 0 = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 0 >= 0 = ok(U51(X1,X2)) U52(ok(X)) = 0 >= 0 = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 0 >= 0 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 1 >= 1 = top(proper(X)) top(ok(X)) = 1 >= 1 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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: active#(length(X)) -> active#(X) active#(s(X)) -> active#(X) active#(U62(X1,X2)) -> active#(X1) active#(U61(X1,X2,X3)) -> active#(X1) active#(U52(X)) -> active#(X) active#(U51(X1,X2)) -> active#(X1) active#(U42(X)) -> active#(X) active#(U41(X1,X2)) -> active#(X1) active#(U31(X)) -> active#(X) active#(U21(X)) -> active#(X) active#(U11(X)) -> active#(X) active#(cons(X1,X2)) -> active#(X1) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + 1, [isNat](x0) = 1, [U61](x0, x1, x2) = x0 + 1, [U52](x0) = x0 + 1, [isNatList](x0) = 1, [U51](x0, x1) = x0 + 1, [U42](x0) = x0 + 1, [isNatIList](x0) = 1, [U41](x0, x1) = x0 + 1, [U31](x0) = x0 + 1, [U21](x0) = x0 + 1, [U11](x0) = x0 + 1, [tt] = 0, [mark](x0) = 1, [cons](x0, x1) = x0 + 1, [0] = 1, [active](x0) = x0, [zeros] = 1 orientation: active#(length(X)) = X + 2 >= X + 1 = active#(X) active#(s(X)) = X + 1 >= X + 1 = active#(X) active#(U62(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U61(X1,X2,X3)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U52(X)) = X + 2 >= X + 1 = active#(X) active#(U51(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U42(X)) = X + 2 >= X + 1 = active#(X) active#(U41(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active#(U31(X)) = X + 2 >= X + 1 = active#(X) active#(U21(X)) = X + 2 >= X + 1 = active#(X) active#(U11(X)) = X + 2 >= X + 1 = active#(X) active#(cons(X1,X2)) = X1 + 2 >= X1 + 1 = active#(X1) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = 1 >= 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = N + 2 >= 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + 1 >= X1 + 1 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = 2 >= 1 = mark(cons(X1,X2)) U11(mark(X)) = 2 >= 1 = mark(U11(X)) U21(mark(X)) = 2 >= 1 = mark(U21(X)) U31(mark(X)) = 2 >= 1 = mark(U31(X)) U41(mark(X1),X2) = 2 >= 1 = mark(U41(X1,X2)) U42(mark(X)) = 2 >= 1 = mark(U42(X)) U51(mark(X1),X2) = 2 >= 1 = mark(U51(X1,X2)) U52(mark(X)) = 2 >= 1 = mark(U52(X)) U61(mark(X1),X2,X3) = 2 >= 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 2 >= 1 = mark(U62(X1,X2)) s(mark(X)) = 1 >= 1 = mark(s(X)) length(mark(X)) = 2 >= 1 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + 1 >= X1 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = 1 >= 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = 1 >= 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 1 >= 1 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: active#(s(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [active#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 0, [s](x0) = x0 + 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 1, [U61](x0, x1, x2) = 1, [U52](x0) = 1, [isNatList](x0) = 1, [U51](x0, x1) = 1, [U42](x0) = 1, [isNatIList](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = 1, [U21](x0) = 1, [U11](x0) = 1, [tt] = 0, [mark](x0) = 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: active#(s(X)) = X + 2 >= X + 1 = active#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = 1 >= 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 1 >= 1 = cons(active(X1),X2) active(U11(X)) = 1 >= 1 = U11(active(X)) active(U21(X)) = 1 >= 1 = U21(active(X)) active(U31(X)) = 1 >= 1 = U31(active(X)) active(U41(X1,X2)) = 1 >= 1 = U41(active(X1),X2) active(U42(X)) = 1 >= 1 = U42(active(X)) active(U51(X1,X2)) = 1 >= 1 = U51(active(X1),X2) active(U52(X)) = 1 >= 1 = U52(active(X)) active(U61(X1,X2,X3)) = 1 >= 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 1 >= 1 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = 1 >= 1 = length(active(X)) cons(mark(X1),X2) = 1 >= 1 = mark(cons(X1,X2)) U11(mark(X)) = 1 >= 1 = mark(U11(X)) U21(mark(X)) = 1 >= 1 = mark(U21(X)) U31(mark(X)) = 1 >= 1 = mark(U31(X)) U41(mark(X1),X2) = 1 >= 1 = mark(U41(X1,X2)) U42(mark(X)) = 1 >= 1 = mark(U42(X)) U51(mark(X1),X2) = 1 >= 1 = mark(U51(X1,X2)) U52(mark(X)) = 1 >= 1 = mark(U52(X)) U61(mark(X1),X2,X3) = 1 >= 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 1 >= 1 = mark(U62(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) length(mark(X)) = 1 >= 1 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 1 >= 1 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 1 >= 1 = U21(proper(X)) proper(U31(X)) = 1 >= 1 = U31(proper(X)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(U42(X)) = 1 >= 1 = U42(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = 1 >= 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = 1 >= 1 = U52(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 1 >= 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 1 >= 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 1 >= 1 = ok(cons(X1,X2)) U11(ok(X)) = 1 >= 1 = ok(U11(X)) U21(ok(X)) = 1 >= 1 = ok(U21(X)) U31(ok(X)) = 1 >= 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = 1 >= 1 = ok(U41(X1,X2)) U42(ok(X)) = 1 >= 1 = ok(U42(X)) isNatIList(ok(X)) = 1 >= 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 1 >= 1 = ok(U51(X1,X2)) U52(ok(X)) = 1 >= 1 = ok(U52(X)) isNatList(ok(X)) = 1 >= 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 1 >= 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 1 >= 1 = ok(U62(X1,X2)) isNat(ok(X)) = 1 >= 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = 1 >= 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatIList#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: isNatIList#(ok(X)) = X + 2 >= X + 1 = isNatIList#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: isNat#(ok(X)) = X + 2 >= X + 1 = isNat#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatList#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: isNatList#(ok(X)) = X + 2 >= X + 1 = isNatList#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X1),ok(X2)) -> cons#(X1,X2) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0, [isNat](x0) = x0, [U61](x0, x1, x2) = x0, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: cons#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X2 + 1 >= X2 + 1 = cons#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = 0 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 0 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = 0 >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X)) = 0 >= 0 = U11(active(X)) active(U21(X)) = 0 >= 0 = U21(active(X)) active(U31(X)) = 0 >= 0 = U31(active(X)) active(U41(X1,X2)) = 0 >= 0 = U41(active(X1),X2) active(U42(X)) = 0 >= 0 = U42(active(X)) active(U51(X1,X2)) = 0 >= 0 = U51(active(X1),X2) active(U52(X)) = 0 >= 0 = U52(active(X)) active(U61(X1,X2,X3)) = 0 >= 0 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 0 >= 0 = U62(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = 0 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = 0 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 0 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X)) = 1 >= 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = 1 >= 1 = U21(proper(X)) proper(U31(X)) = 1 >= 1 = U31(proper(X)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(U42(X)) = 1 >= 1 = U42(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = 1 >= 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = 1 >= 1 = U52(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 1 >= 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 1 >= 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 1, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0 + x1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + x2, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: cons#(mark(X1),X2) = X1 + 1 >= X1 = cons#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + N + 1 >= L + 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + N + 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 0 >= 0 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(U31(X)) = 0 >= 0 = U31(proper(X)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(U42(X)) = 0 >= 0 = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = 0 >= 0 = U51(proper(X1),proper(X2)) proper(U52(X)) = 0 >= 0 = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 0 >= 0 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 0 >= 0 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X)) = 0 >= 0 = ok(U11(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X)) = 0 >= 0 = ok(U31(X)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) U42(ok(X)) = 0 >= 0 = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 0 >= 0 = ok(U51(X1,X2)) U52(ok(X)) = 0 >= 0 = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 0 >= 0 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X)) -> U11#(X) U11#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + x1 + 1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + 1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U11#(ok(X)) = X + 1 >= X + 1 = U11#(X) U11#(mark(X)) = X + 2 >= X + 1 = U11#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + 2 >= L + 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 2 >= L + 2 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 3 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 2 >= L + 2 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 2 >= X1 + X2 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 2 >= X + 2 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 >= X1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 >= X1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 >= X1 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 >= X1 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: U11#(ok(X)) = X + 2 >= X + 1 = U11#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X)) -> U21#(X) U21#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + x1 + 1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + 1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U21#(ok(X)) = X + 1 >= X + 1 = U21#(X) U21#(mark(X)) = X + 2 >= X + 1 = U21#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + 2 >= L + 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 2 >= L + 2 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 3 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 2 >= L + 2 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 2 >= X1 + X2 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 2 >= X + 2 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 >= X1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 >= X1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 >= X1 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 >= X1 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: U21#(ok(X)) = X + 2 >= X + 1 = U21#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X)) -> U31#(X) U31#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + x1 + 1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + 1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U31#(ok(X)) = X + 1 >= X + 1 = U31#(X) U31#(mark(X)) = X + 2 >= X + 1 = U31#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + 2 >= L + 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 2 >= L + 2 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 3 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 2 >= L + 2 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 2 >= X1 + X2 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 2 >= X + 2 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 >= X1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 >= X1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 >= X1 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 >= X1 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: U31#(ok(X)) = X + 2 >= X + 1 = U31#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X1),ok(X2)) -> U41#(X1,X2) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0, [isNat](x0) = x0, [U61](x0, x1, x2) = x0, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U41#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U41#(X1,X2) U41#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U41#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = 0 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 0 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = 0 >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X)) = 0 >= 0 = U11(active(X)) active(U21(X)) = 0 >= 0 = U21(active(X)) active(U31(X)) = 0 >= 0 = U31(active(X)) active(U41(X1,X2)) = 0 >= 0 = U41(active(X1),X2) active(U42(X)) = 0 >= 0 = U42(active(X)) active(U51(X1,X2)) = 0 >= 0 = U51(active(X1),X2) active(U52(X)) = 0 >= 0 = U52(active(X)) active(U61(X1,X2,X3)) = 0 >= 0 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 0 >= 0 = U62(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = 0 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = 0 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 0 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X)) = 1 >= 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = 1 >= 1 = U21(proper(X)) proper(U31(X)) = 1 >= 1 = U31(proper(X)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(U42(X)) = 1 >= 1 = U42(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = 1 >= 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = 1 >= 1 = U52(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 1 >= 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 1 >= 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 1, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0 + x1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + x2, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U41#(mark(X1),X2) = X1 + 1 >= X1 = U41#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + N + 1 >= L + 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + N + 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 0 >= 0 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(U31(X)) = 0 >= 0 = U31(proper(X)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(U42(X)) = 0 >= 0 = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = 0 >= 0 = U51(proper(X1),proper(X2)) proper(U52(X)) = 0 >= 0 = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 0 >= 0 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 0 >= 0 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X)) = 0 >= 0 = ok(U11(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X)) = 0 >= 0 = ok(U31(X)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) U42(ok(X)) = 0 >= 0 = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 0 >= 0 = ok(U51(X1,X2)) U52(ok(X)) = 0 >= 0 = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 0 >= 0 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X)) -> U42#(X) U42#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U42#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + x1 + 1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + 1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U42#(ok(X)) = X + 1 >= X + 1 = U42#(X) U42#(mark(X)) = X + 2 >= X + 1 = U42#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + 2 >= L + 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 2 >= L + 2 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 3 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 2 >= L + 2 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 2 >= X1 + X2 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 2 >= X + 2 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 >= X1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 >= X1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 >= X1 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 >= X1 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U42#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: U42#(ok(X)) = X + 2 >= X + 1 = U42#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X1),ok(X2)) -> U51#(X1,X2) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0, [isNat](x0) = x0, [U61](x0, x1, x2) = x0, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U51#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U51#(X1,X2) U51#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U51#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = 0 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 0 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = 0 >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X)) = 0 >= 0 = U11(active(X)) active(U21(X)) = 0 >= 0 = U21(active(X)) active(U31(X)) = 0 >= 0 = U31(active(X)) active(U41(X1,X2)) = 0 >= 0 = U41(active(X1),X2) active(U42(X)) = 0 >= 0 = U42(active(X)) active(U51(X1,X2)) = 0 >= 0 = U51(active(X1),X2) active(U52(X)) = 0 >= 0 = U52(active(X)) active(U61(X1,X2,X3)) = 0 >= 0 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 0 >= 0 = U62(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = 0 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = 0 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 0 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X)) = 1 >= 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = 1 >= 1 = U21(proper(X)) proper(U31(X)) = 1 >= 1 = U31(proper(X)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(U42(X)) = 1 >= 1 = U42(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = 1 >= 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = 1 >= 1 = U52(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 1 >= 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 1 >= 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 1, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0 + x1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + x2, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U51#(mark(X1),X2) = X1 + 1 >= X1 = U51#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + N + 1 >= L + 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + N + 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 0 >= 0 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(U31(X)) = 0 >= 0 = U31(proper(X)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(U42(X)) = 0 >= 0 = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = 0 >= 0 = U51(proper(X1),proper(X2)) proper(U52(X)) = 0 >= 0 = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 0 >= 0 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 0 >= 0 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X)) = 0 >= 0 = ok(U11(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X)) = 0 >= 0 = ok(U31(X)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) U42(ok(X)) = 0 >= 0 = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 0 >= 0 = ok(U51(X1,X2)) U52(ok(X)) = 0 >= 0 = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 0 >= 0 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X)) -> U52#(X) U52#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U52#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + x1 + 1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + 1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U52#(ok(X)) = X + 1 >= X + 1 = U52#(X) U52#(mark(X)) = X + 2 >= X + 1 = U52#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + 2 >= L + 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 2 >= L + 2 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 3 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 2 >= L + 2 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 2 >= X1 + X2 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 2 >= X + 2 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 >= X1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 >= X1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 >= X1 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 >= X1 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U52#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: U52#(ok(X)) = X + 2 >= X + 1 = U52#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X1),ok(X2),ok(X3)) -> U61#(X1,X2,X3) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1, x2) = x2 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0 + 1, [length](x0) = x0, [U62](x0, x1) = x0, [isNat](x0) = x0, [U61](x0, x1, x2) = x0, [U52](x0) = x0 + 1, [isNatList](x0) = x0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0 + 1, [U11](x0) = x0 + 1, [tt] = 0, [mark](x0) = 1, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U61#(ok(X1),ok(X2),ok(X3)) = X3 + 2 >= X3 + 1 = U61#(X1,X2,X3) U61#(mark(X1),X2,X3) = X3 + 1 >= X3 + 1 = U61#(X1,X2,X3) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 2 >= 1 = mark(tt()) active(U21(tt())) = 2 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = V2 + 2 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 2 >= 1 = mark(tt()) active(U61(tt(),L,N)) = 1 >= 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 1 >= 1 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = V1 + 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 + 2 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = V + 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = V1 + 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = N + 1 >= 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(active(X1),X2) active(U11(X)) = X + 2 >= X + 2 = U11(active(X)) active(U21(X)) = X + 2 >= X + 2 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X2 + 2 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 2 >= X + 2 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + 1 >= X1 + 1 = U62(active(X1),X2) active(s(X)) = X + 2 >= X + 2 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = 1 >= 1 = mark(cons(X1,X2)) U11(mark(X)) = 2 >= 1 = mark(U11(X)) U21(mark(X)) = 2 >= 1 = mark(U21(X)) U31(mark(X)) = 1 >= 1 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 1 = mark(U41(X1,X2)) U42(mark(X)) = 1 >= 1 = mark(U42(X)) U51(mark(X1),X2) = 1 >= 1 = mark(U51(X1,X2)) U52(mark(X)) = 2 >= 1 = mark(U52(X)) U61(mark(X1),X2,X3) = 1 >= 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 1 >= 1 = mark(U62(X1,X2)) s(mark(X)) = 2 >= 1 = mark(s(X)) length(mark(X)) = 1 >= 1 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X)) = X + 2 >= X + 2 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 2 >= X + 2 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 2 >= X + 2 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + 1 >= X1 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + 1 >= X1 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 2 >= X + 2 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 2 >= X + 2 = ok(U11(X)) U21(ok(X)) = X + 2 >= X + 2 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 2 >= X + 2 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 2 >= X + 2 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1, x2) = x0 + 1, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0 + x1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U61#(mark(X1),X2,X3) = X1 + 2 >= X1 + 1 = U61#(X1,X2,X3) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + 1 >= L + 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 1 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 0 >= 0 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(U31(X)) = 0 >= 0 = U31(proper(X)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(U42(X)) = 0 >= 0 = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = 0 >= 0 = U51(proper(X1),proper(X2)) proper(U52(X)) = 0 >= 0 = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 0 >= 0 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 0 >= 0 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X)) = 0 >= 0 = ok(U11(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X)) = 0 >= 0 = ok(U31(X)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) U42(ok(X)) = 0 >= 0 = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 0 >= 0 = ok(U51(X1,X2)) U52(ok(X)) = 0 >= 0 = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 0 >= 0 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X1),ok(X2)) -> U62#(X1,X2) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U62#](x0, x1) = x1 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0, [isNat](x0) = x0, [U61](x0, x1, x2) = x0, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: U62#(ok(X1),ok(X2)) = X2 + 2 >= X2 + 1 = U62#(X1,X2) U62#(mark(X1),X2) = X2 + 1 >= X2 + 1 = U62#(X1,X2) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = 0 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = 0 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = 0 >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 0 >= 0 = mark(s(length(L))) active(isNat(0())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 0 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 0 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = 0 >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 0 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 0 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = 0 >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = 0 >= 0 = cons(active(X1),X2) active(U11(X)) = 0 >= 0 = U11(active(X)) active(U21(X)) = 0 >= 0 = U21(active(X)) active(U31(X)) = 0 >= 0 = U31(active(X)) active(U41(X1,X2)) = 0 >= 0 = U41(active(X1),X2) active(U42(X)) = 0 >= 0 = U42(active(X)) active(U51(X1,X2)) = 0 >= 0 = U51(active(X1),X2) active(U52(X)) = 0 >= 0 = U52(active(X)) active(U61(X1,X2,X3)) = 0 >= 0 = U61(active(X1),X2,X3) active(U62(X1,X2)) = 0 >= 0 = U62(active(X1),X2) active(s(X)) = 0 >= 0 = s(active(X)) active(length(X)) = 0 >= 0 = length(active(X)) cons(mark(X1),X2) = 0 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = 0 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = 0 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = 0 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = 0 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = 1 >= 1 = cons(proper(X1),proper(X2)) proper(0()) = 1 >= 1 = ok(0()) proper(U11(X)) = 1 >= 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = 1 >= 1 = U21(proper(X)) proper(U31(X)) = 1 >= 1 = U31(proper(X)) proper(U41(X1,X2)) = 1 >= 1 = U41(proper(X1),proper(X2)) proper(U42(X)) = 1 >= 1 = U42(proper(X)) proper(isNatIList(X)) = 1 >= 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = 1 >= 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = 1 >= 1 = U52(proper(X)) proper(isNatList(X)) = 1 >= 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 1 >= 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 1 >= 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 1 >= 1 = isNat(proper(X)) proper(s(X)) = 1 >= 1 = s(proper(X)) proper(length(X)) = 1 >= 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + 1 >= X1 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + 1 >= X1 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [U62#](x0, x1) = x0, [top](x0) = 0, [ok](x0) = 0, [proper](x0) = 0, [nil] = 1, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x0 + x1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + x2, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U62#(mark(X1),X2) = X1 + 1 >= X1 = U62#(X1,X2) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + N + 1 >= L + 1 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 1 >= L + 1 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 2 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + N + 1 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 1 >= X + 1 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 1 >= X + 1 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = 0 >= 0 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = 0 >= 0 = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = 0 >= 0 = U21(proper(X)) proper(U31(X)) = 0 >= 0 = U31(proper(X)) proper(U41(X1,X2)) = 0 >= 0 = U41(proper(X1),proper(X2)) proper(U42(X)) = 0 >= 0 = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = 0 >= 0 = U51(proper(X1),proper(X2)) proper(U52(X)) = 0 >= 0 = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = 0 >= 0 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = 0 >= 0 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = 0 >= 0 = s(proper(X)) proper(length(X)) = 0 >= 0 = length(proper(X)) proper(nil()) = 0 >= 0 = ok(nil()) cons(ok(X1),ok(X2)) = 0 >= 0 = ok(cons(X1,X2)) U11(ok(X)) = 0 >= 0 = ok(U11(X)) U21(ok(X)) = 0 >= 0 = ok(U21(X)) U31(ok(X)) = 0 >= 0 = ok(U31(X)) U41(ok(X1),ok(X2)) = 0 >= 0 = ok(U41(X1,X2)) U42(ok(X)) = 0 >= 0 = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = 0 >= 0 = ok(U51(X1,X2)) U52(ok(X)) = 0 >= 0 = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = 0 >= 0 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = 0 >= 0 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = 0 >= 0 = ok(s(X)) length(ok(X)) = 0 >= 0 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X)) -> s#(X) s#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + x1 + 1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + 1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: s#(ok(X)) = X + 1 >= X + 1 = s#(X) s#(mark(X)) = X + 2 >= X + 1 = s#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + 2 >= L + 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 2 >= L + 2 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 3 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 2 >= L + 2 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 2 >= X1 + X2 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 2 >= X + 2 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 >= X1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 >= X1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 >= X1 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 >= X1 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: s#(ok(X)) -> s#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: s#(ok(X)) = X + 2 >= X + 1 = s#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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#(ok(X)) -> length#(X) length#(mark(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)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0, [proper](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x0 + x1 + 1, [isNat](x0) = 0, [U61](x0, x1, x2) = x0 + x1 + 1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = x0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: length#(ok(X)) = X + 1 >= X + 1 = length#(X) length#(mark(X)) = X + 2 >= X + 1 = length#(X) active(zeros()) = 1 >= 1 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 1 >= 1 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = mark(U42(isNatIList(V2))) active(U42(tt())) = 1 >= 1 = mark(tt()) active(U51(tt(),V2)) = 1 >= 1 = mark(U52(isNatList(V2))) active(U52(tt())) = 1 >= 1 = mark(tt()) active(U61(tt(),L,N)) = L + 2 >= L + 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L + 2 >= L + 2 = mark(s(length(L))) active(isNat(0())) = 1 >= 1 = mark(tt()) active(isNat(length(V1))) = 1 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 1 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 1 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 3 >= 1 = mark(0()) active(length(cons(N,L))) = L + N + 2 >= L + 2 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(active(X1),X2) active(U11(X)) = X + 1 >= X + 1 = U11(active(X)) active(U21(X)) = X + 1 >= X + 1 = U21(active(X)) active(U31(X)) = X + 1 >= X + 1 = U31(active(X)) active(U41(X1,X2)) = X1 + 1 >= X1 + 1 = U41(active(X1),X2) active(U42(X)) = X + 1 >= X + 1 = U42(active(X)) active(U51(X1,X2)) = X1 + 1 >= X1 + 1 = U51(active(X1),X2) active(U52(X)) = X + 1 >= X + 1 = U52(active(X)) active(U61(X1,X2,X3)) = X1 + X2 + 2 >= X1 + X2 + 2 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = U62(active(X1),X2) active(s(X)) = X + 1 >= X + 1 = s(active(X)) active(length(X)) = X + 2 >= X + 2 = length(active(X)) cons(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = mark(cons(X1,X2)) U11(mark(X)) = X + 1 >= X + 1 = mark(U11(X)) U21(mark(X)) = X + 1 >= X + 1 = mark(U21(X)) U31(mark(X)) = X + 1 >= X + 1 = mark(U31(X)) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U41(X1,X2)) U42(mark(X)) = X + 1 >= X + 1 = mark(U42(X)) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = mark(U51(X1,X2)) U52(mark(X)) = X + 1 >= X + 1 = mark(U52(X)) U61(mark(X1),X2,X3) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 2 = mark(U62(X1,X2)) s(mark(X)) = X + 1 >= X + 1 = mark(s(X)) length(mark(X)) = X + 2 >= X + 2 = mark(length(X)) proper(zeros()) = 0 >= 0 = ok(zeros()) proper(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(proper(X1),proper(X2)) proper(0()) = 0 >= 0 = ok(0()) proper(U11(X)) = X >= X = U11(proper(X)) proper(tt()) = 0 >= 0 = ok(tt()) proper(U21(X)) = X >= X = U21(proper(X)) proper(U31(X)) = X >= X = U31(proper(X)) proper(U41(X1,X2)) = X1 >= X1 = U41(proper(X1),proper(X2)) proper(U42(X)) = X >= X = U42(proper(X)) proper(isNatIList(X)) = 0 >= 0 = isNatIList(proper(X)) proper(U51(X1,X2)) = X1 >= X1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X >= X = U52(proper(X)) proper(isNatList(X)) = 0 >= 0 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = 0 >= 0 = isNat(proper(X)) proper(s(X)) = X >= X = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X1 + X2 >= X1 + X2 = ok(cons(X1,X2)) U11(ok(X)) = X >= X = ok(U11(X)) U21(ok(X)) = X >= X = ok(U21(X)) U31(ok(X)) = X >= X = ok(U31(X)) U41(ok(X1),ok(X2)) = X1 >= X1 = ok(U41(X1,X2)) U42(ok(X)) = X >= X = ok(U42(X)) isNatIList(ok(X)) = 0 >= 0 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X1 >= X1 = ok(U51(X1,X2)) U52(ok(X)) = X >= X = ok(U52(X)) isNatList(ok(X)) = 0 >= 0 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = 0 >= 0 = ok(isNat(X)) s(ok(X)) = X >= X = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: length#(ok(X)) -> length#(X) TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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)) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [top](x0) = 0, [ok](x0) = x0 + 1, [proper](x0) = x0 + 1, [nil] = 0, [s](x0) = x0, [length](x0) = x0, [U62](x0, x1) = x1, [isNat](x0) = x0, [U61](x0, x1, x2) = x2, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x1, [U42](x0) = x0, [isNatIList](x0) = x0, [U41](x0, x1) = x1 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = x1, [0] = 1, [active](x0) = x0, [zeros] = 0 orientation: length#(ok(X)) = X + 2 >= X + 1 = length#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 0 >= 0 = mark(tt()) active(U21(tt())) = 0 >= 0 = mark(tt()) active(U31(tt())) = 0 >= 0 = mark(tt()) active(U41(tt(),V2)) = V2 + 1 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= 0 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = mark(tt()) active(U61(tt(),L,N)) = N >= 0 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= 0 = mark(s(length(L))) active(isNat(0())) = 1 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= 0 = mark(U21(isNat(V1))) active(isNatIList(V)) = V >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 0 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = V2 >= 0 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 0 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V2 >= 0 = mark(U51(isNat(V1),V2)) active(length(nil())) = 0 >= 0 = mark(0()) active(length(cons(N,L))) = L >= 0 = mark(U61(isNatList(L),L,N)) active(cons(X1,X2)) = X2 >= X2 = cons(active(X1),X2) active(U11(X)) = X >= X = U11(active(X)) active(U21(X)) = X >= X = U21(active(X)) active(U31(X)) = X >= X = U31(active(X)) active(U41(X1,X2)) = X2 + 1 >= X2 + 1 = U41(active(X1),X2) active(U42(X)) = X >= X = U42(active(X)) active(U51(X1,X2)) = X2 >= X2 = U51(active(X1),X2) active(U52(X)) = X >= X = U52(active(X)) active(U61(X1,X2,X3)) = X3 >= X3 = U61(active(X1),X2,X3) active(U62(X1,X2)) = X2 >= X2 = U62(active(X1),X2) active(s(X)) = X >= X = s(active(X)) active(length(X)) = X >= X = length(active(X)) cons(mark(X1),X2) = X2 >= 0 = mark(cons(X1,X2)) U11(mark(X)) = 0 >= 0 = mark(U11(X)) U21(mark(X)) = 0 >= 0 = mark(U21(X)) U31(mark(X)) = 0 >= 0 = mark(U31(X)) U41(mark(X1),X2) = X2 + 1 >= 0 = mark(U41(X1,X2)) U42(mark(X)) = 0 >= 0 = mark(U42(X)) U51(mark(X1),X2) = X2 >= 0 = mark(U51(X1,X2)) U52(mark(X)) = 0 >= 0 = mark(U52(X)) U61(mark(X1),X2,X3) = X3 >= 0 = mark(U61(X1,X2,X3)) U62(mark(X1),X2) = X2 >= 0 = mark(U62(X1,X2)) s(mark(X)) = 0 >= 0 = mark(s(X)) length(mark(X)) = 0 >= 0 = mark(length(X)) proper(zeros()) = 1 >= 1 = ok(zeros()) proper(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(proper(X1),proper(X2)) proper(0()) = 2 >= 2 = ok(0()) proper(U11(X)) = X + 1 >= X + 1 = U11(proper(X)) proper(tt()) = 1 >= 1 = ok(tt()) proper(U21(X)) = X + 1 >= X + 1 = U21(proper(X)) proper(U31(X)) = X + 1 >= X + 1 = U31(proper(X)) proper(U41(X1,X2)) = X2 + 2 >= X2 + 2 = U41(proper(X1),proper(X2)) proper(U42(X)) = X + 1 >= X + 1 = U42(proper(X)) proper(isNatIList(X)) = X + 1 >= X + 1 = isNatIList(proper(X)) proper(U51(X1,X2)) = X2 + 1 >= X2 + 1 = U51(proper(X1),proper(X2)) proper(U52(X)) = X + 1 >= X + 1 = U52(proper(X)) proper(isNatList(X)) = X + 1 >= X + 1 = isNatList(proper(X)) proper(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = U61(proper(X1),proper(X2),proper(X3)) proper(U62(X1,X2)) = X2 + 1 >= X2 + 1 = U62(proper(X1),proper(X2)) proper(isNat(X)) = X + 1 >= X + 1 = isNat(proper(X)) proper(s(X)) = X + 1 >= X + 1 = s(proper(X)) proper(length(X)) = X + 1 >= X + 1 = length(proper(X)) proper(nil()) = 1 >= 1 = ok(nil()) cons(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(cons(X1,X2)) U11(ok(X)) = X + 1 >= X + 1 = ok(U11(X)) U21(ok(X)) = X + 1 >= X + 1 = ok(U21(X)) U31(ok(X)) = X + 1 >= X + 1 = ok(U31(X)) U41(ok(X1),ok(X2)) = X2 + 2 >= X2 + 2 = ok(U41(X1,X2)) U42(ok(X)) = X + 1 >= X + 1 = ok(U42(X)) isNatIList(ok(X)) = X + 1 >= X + 1 = ok(isNatIList(X)) U51(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U51(X1,X2)) U52(ok(X)) = X + 1 >= X + 1 = ok(U52(X)) isNatList(ok(X)) = X + 1 >= X + 1 = ok(isNatList(X)) U61(ok(X1),ok(X2),ok(X3)) = X3 + 1 >= X3 + 1 = ok(U61(X1,X2,X3)) U62(ok(X1),ok(X2)) = X2 + 1 >= X2 + 1 = ok(U62(X1,X2)) isNat(ok(X)) = X + 1 >= X + 1 = ok(isNat(X)) s(ok(X)) = X + 1 >= X + 1 = ok(s(X)) length(ok(X)) = X + 1 >= X + 1 = ok(length(X)) top(mark(X)) = 0 >= 0 = top(proper(X)) top(ok(X)) = 0 >= 0 = top(active(X)) problem: DPs: TRS: active(zeros()) -> mark(cons(0(),zeros())) active(U11(tt())) -> 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