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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Proof: DP Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) 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)) -> isNatIList#(V2) active#(U41(tt(),V2)) -> U42#(isNatIList(V2)) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) active#(U42(tt())) -> mark#(tt()) active#(U51(tt(),V2)) -> isNatList#(V2) active#(U51(tt(),V2)) -> U52#(isNatList(V2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) active#(U52(tt())) -> mark#(tt()) active#(U61(tt(),L,N)) -> isNat#(N) active#(U61(tt(),L,N)) -> U62#(isNat(N),L) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) active#(U62(tt(),L)) -> length#(L) active#(U62(tt(),L)) -> s#(length(L)) active#(U62(tt(),L)) -> mark#(s(length(L))) active#(isNat(0())) -> mark#(tt()) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(length(V1))) -> U11#(isNatList(V1)) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> U21#(isNat(V1)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(isNatIList(V)) -> isNatList#(V) active#(isNatIList(V)) -> U31#(isNatList(V)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(V1,V2))) -> isNat#(V1) active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> isNat#(V1) active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> U61#(isNatList(L),L,N) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> U11#(mark(X)) mark#(U11(X)) -> active#(U11(mark(X))) mark#(tt()) -> active#(tt()) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> U21#(mark(X)) mark#(U21(X)) -> active#(U21(mark(X))) mark#(U31(X)) -> mark#(X) mark#(U31(X)) -> U31#(mark(X)) mark#(U31(X)) -> active#(U31(mark(X))) mark#(U41(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> U41#(mark(X1),X2) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) mark#(U42(X)) -> mark#(X) mark#(U42(X)) -> U42#(mark(X)) mark#(U42(X)) -> active#(U42(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> U51#(mark(X1),X2) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) mark#(U52(X)) -> mark#(X) mark#(U52(X)) -> U52#(mark(X)) mark#(U52(X)) -> active#(U52(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> U62#(mark(X1),X2) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(nil()) -> active#(nil()) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) U11#(mark(X)) -> U11#(X) U11#(active(X)) -> U11#(X) U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) U31#(mark(X)) -> U31#(X) U31#(active(X)) -> U31#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) U41#(active(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U42#(mark(X)) -> U42#(X) U42#(active(X)) -> U42#(X) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U51#(active(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U52#(mark(X)) -> U52#(X) U52#(active(X)) -> U52#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) U62#(active(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) EDG Processor: DPs: active#(zeros()) -> cons#(0(),zeros()) 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)) -> isNatIList#(V2) active#(U41(tt(),V2)) -> U42#(isNatIList(V2)) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) active#(U42(tt())) -> mark#(tt()) active#(U51(tt(),V2)) -> isNatList#(V2) active#(U51(tt(),V2)) -> U52#(isNatList(V2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) active#(U52(tt())) -> mark#(tt()) active#(U61(tt(),L,N)) -> isNat#(N) active#(U61(tt(),L,N)) -> U62#(isNat(N),L) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) active#(U62(tt(),L)) -> length#(L) active#(U62(tt(),L)) -> s#(length(L)) active#(U62(tt(),L)) -> mark#(s(length(L))) active#(isNat(0())) -> mark#(tt()) active#(isNat(length(V1))) -> isNatList#(V1) active#(isNat(length(V1))) -> U11#(isNatList(V1)) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) active#(isNat(s(V1))) -> isNat#(V1) active#(isNat(s(V1))) -> U21#(isNat(V1)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) active#(isNatIList(V)) -> isNatList#(V) active#(isNatIList(V)) -> U31#(isNatList(V)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) active#(isNatIList(zeros())) -> mark#(tt()) active#(isNatIList(cons(V1,V2))) -> isNat#(V1) active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) active#(isNatList(nil())) -> mark#(tt()) active#(isNatList(cons(V1,V2))) -> isNat#(V1) active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) active#(length(nil())) -> mark#(0()) active#(length(cons(N,L))) -> isNatList#(L) active#(length(cons(N,L))) -> U61#(isNatList(L),L,N) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) mark#(zeros()) -> active#(zeros()) mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> cons#(mark(X1),X2) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(0()) -> active#(0()) mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> U11#(mark(X)) mark#(U11(X)) -> active#(U11(mark(X))) mark#(tt()) -> active#(tt()) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> U21#(mark(X)) mark#(U21(X)) -> active#(U21(mark(X))) mark#(U31(X)) -> mark#(X) mark#(U31(X)) -> U31#(mark(X)) mark#(U31(X)) -> active#(U31(mark(X))) mark#(U41(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> U41#(mark(X1),X2) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) mark#(U42(X)) -> mark#(X) mark#(U42(X)) -> U42#(mark(X)) mark#(U42(X)) -> active#(U42(mark(X))) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> U51#(mark(X1),X2) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) mark#(U52(X)) -> mark#(X) mark#(U52(X)) -> U52#(mark(X)) mark#(U52(X)) -> active#(U52(mark(X))) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> U62#(mark(X1),X2) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) mark#(isNat(X)) -> active#(isNat(X)) mark#(s(X)) -> mark#(X) mark#(s(X)) -> s#(mark(X)) mark#(s(X)) -> active#(s(mark(X))) mark#(length(X)) -> mark#(X) mark#(length(X)) -> length#(mark(X)) mark#(length(X)) -> active#(length(mark(X))) mark#(nil()) -> active#(nil()) cons#(mark(X1),X2) -> cons#(X1,X2) cons#(X1,mark(X2)) -> cons#(X1,X2) cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) U11#(mark(X)) -> U11#(X) U11#(active(X)) -> U11#(X) U21#(mark(X)) -> U21#(X) U21#(active(X)) -> U21#(X) U31#(mark(X)) -> U31#(X) U31#(active(X)) -> U31#(X) U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,mark(X2)) -> U41#(X1,X2) U41#(active(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U42#(mark(X)) -> U42#(X) U42#(active(X)) -> U42#(X) isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(X)) -> isNatIList#(X) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U51#(active(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U52#(mark(X)) -> U52#(X) U52#(active(X)) -> U52#(X) isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(X)) -> isNatList#(X) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,mark(X2)) -> U62#(X1,X2) U62#(active(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) isNat#(mark(X)) -> isNat#(X) isNat#(active(X)) -> isNat#(X) s#(mark(X)) -> s#(X) s#(active(X)) -> s#(X) length#(mark(X)) -> length#(X) length#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) graph: ... SCC Processor: #sccs: 16 #rules: 81 #arcs: 1590/15876 DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) mark#(length(X)) -> mark#(X) mark#(s(X)) -> active#(s(mark(X))) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> active#(U52(mark(X))) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(U42(X)) -> active#(U42(mark(X))) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U41(X1,X2)) -> mark#(X1) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U31(X)) -> mark#(X) mark#(U21(X)) -> active#(U21(mark(X))) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> active#(U11(mark(X))) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [nil] = 0, [s](x0) = 0, [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) = 0, [U11](x0) = 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(length(cons(N,L))) = 1 >= 1 = mark#(U61(isNatList(L),L,N)) mark#(length(X)) = 1 >= 1 = mark#(X) mark#(s(X)) = 1 >= 0 = active#(s(mark(X))) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(U52(X)) = 1 >= 1 = active#(U52(mark(X))) active#(isNat(length(V1))) = 1 >= 1 = mark#(U11(isNatList(V1))) mark#(U52(X)) = 1 >= 1 = mark#(X) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2)) = 1 >= 1 = mark#(X1) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) mark#(U42(X)) = 1 >= 1 = active#(U42(mark(X))) active#(U61(tt(),L,N)) = 1 >= 1 = mark#(U62(isNat(N),L)) mark#(U42(X)) = 1 >= 1 = mark#(X) mark#(U41(X1,X2)) = 1 >= 1 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 1 >= 1 = mark#(U52(isNatList(V2))) mark#(U41(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X)) = 1 >= 1 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 1 >= 1 = mark#(U42(isNatIList(V2))) mark#(U31(X)) = 1 >= 1 = mark#(X) mark#(U21(X)) = 1 >= 0 = active#(U21(mark(X))) mark#(U21(X)) = 1 >= 1 = mark#(X) mark#(U11(X)) = 1 >= 1 = active#(U11(mark(X))) mark#(U11(X)) = 1 >= 1 = mark#(X) mark#(cons(X1,X2)) = 1 >= 1 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) 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)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = 0 >= 0 = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = 0 >= 0 = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X)) = 0 >= 0 = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 1 >= 1 = U11(X) U11(active(X)) = 1 >= 1 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 1 >= 1 = U31(X) U31(active(X)) = 1 >= 1 = U31(X) U41(mark(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,mark(X2)) = 1 >= 1 = U41(X1,X2) U41(active(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,active(X2)) = 1 >= 1 = U41(X1,X2) U42(mark(X)) = 1 >= 1 = U42(X) U42(active(X)) = 1 >= 1 = U42(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X)) = 1 >= 1 = U52(X) U52(active(X)) = 1 >= 1 = U52(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) mark#(length(X)) -> mark#(X) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> active#(U52(mark(X))) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(U42(X)) -> active#(U42(mark(X))) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U41(X1,X2)) -> mark#(X1) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U31(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> active#(U11(mark(X))) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [nil] = 0, [s](x0) = 0, [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) = 0, [U11](x0) = 1, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(length(cons(N,L))) = 1 >= 1 = mark#(U61(isNatList(L),L,N)) mark#(length(X)) = 1 >= 1 = mark#(X) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(U52(X)) = 1 >= 1 = active#(U52(mark(X))) active#(isNat(length(V1))) = 1 >= 1 = mark#(U11(isNatList(V1))) mark#(U52(X)) = 1 >= 1 = mark#(X) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2)) = 1 >= 1 = mark#(X1) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) mark#(U42(X)) = 1 >= 1 = active#(U42(mark(X))) active#(U61(tt(),L,N)) = 1 >= 1 = mark#(U62(isNat(N),L)) mark#(U42(X)) = 1 >= 1 = mark#(X) mark#(U41(X1,X2)) = 1 >= 1 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 1 >= 1 = mark#(U52(isNatList(V2))) mark#(U41(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X)) = 1 >= 1 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 1 >= 1 = mark#(U42(isNatIList(V2))) mark#(U31(X)) = 1 >= 1 = mark#(X) mark#(U21(X)) = 1 >= 1 = mark#(X) mark#(U11(X)) = 1 >= 1 = active#(U11(mark(X))) mark#(U11(X)) = 1 >= 1 = mark#(X) mark#(cons(X1,X2)) = 1 >= 0 = active#(cons(mark(X1),X2)) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) 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)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = 0 >= 0 = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = 0 >= 0 = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X)) = 0 >= 0 = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X)) = 1 >= 1 = U11(X) U11(active(X)) = 1 >= 1 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 1 >= 1 = U31(X) U31(active(X)) = 1 >= 1 = U31(X) U41(mark(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,mark(X2)) = 1 >= 1 = U41(X1,X2) U41(active(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,active(X2)) = 1 >= 1 = U41(X1,X2) U42(mark(X)) = 1 >= 1 = U42(X) U42(active(X)) = 1 >= 1 = U42(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X)) = 1 >= 1 = U52(X) U52(active(X)) = 1 >= 1 = U52(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) mark#(length(X)) -> mark#(X) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> active#(U52(mark(X))) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) mark#(U42(X)) -> active#(U42(mark(X))) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U41(X1,X2)) -> mark#(X1) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U31(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> active#(U11(mark(X))) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [nil] = 0, [s](x0) = 0, [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) = 0, [isNatIList](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = 1, [U21](x0) = 1, [U11](x0) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(length(cons(N,L))) = 1 >= 1 = mark#(U61(isNatList(L),L,N)) mark#(length(X)) = 1 >= 1 = mark#(X) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) = 1 >= 1 = mark#(X1) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(U52(X)) = 1 >= 1 = active#(U52(mark(X))) active#(isNat(length(V1))) = 1 >= 1 = mark#(U11(isNatList(V1))) mark#(U52(X)) = 1 >= 1 = mark#(X) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2)) = 1 >= 1 = mark#(X1) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) mark#(U42(X)) = 1 >= 0 = active#(U42(mark(X))) active#(U61(tt(),L,N)) = 1 >= 1 = mark#(U62(isNat(N),L)) mark#(U42(X)) = 1 >= 1 = mark#(X) mark#(U41(X1,X2)) = 1 >= 1 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 1 >= 1 = mark#(U52(isNatList(V2))) mark#(U41(X1,X2)) = 1 >= 1 = mark#(X1) mark#(U31(X)) = 1 >= 1 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 1 >= 1 = mark#(U42(isNatIList(V2))) mark#(U31(X)) = 1 >= 1 = mark#(X) mark#(U21(X)) = 1 >= 1 = mark#(X) mark#(U11(X)) = 1 >= 0 = active#(U11(mark(X))) mark#(U11(X)) = 1 >= 1 = mark#(X) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) mark#(zeros()) = 1 >= 1 = active#(zeros()) active#(zeros()) = 1 >= 1 = mark#(cons(0(),zeros())) 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)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = 0 >= 0 = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = 0 >= 0 = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X)) = 0 >= 0 = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 1 >= 1 = U21(X) U21(active(X)) = 1 >= 1 = U21(X) U31(mark(X)) = 1 >= 1 = U31(X) U31(active(X)) = 1 >= 1 = U31(X) U41(mark(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,mark(X2)) = 1 >= 1 = U41(X1,X2) U41(active(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,active(X2)) = 1 >= 1 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X)) = 1 >= 1 = U52(X) U52(active(X)) = 1 >= 1 = U52(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) mark#(length(X)) -> mark#(X) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> active#(U52(mark(X))) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U41(X1,X2)) -> mark#(X1) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U31(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [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) = 1, [U41](x0, x1) = x0 + 1, [U31](x0) = x0 + 1, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X >= X = active#(length(mark(X))) active#(length(cons(N,L))) = L + N >= L = mark#(U61(isNatList(L),L,N)) mark#(length(X)) = X >= X = mark#(X) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 >= X1 + X2 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) = X1 + X2 >= X1 = mark#(X1) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) mark#(U52(X)) = X >= X = active#(U52(mark(X))) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatList(V1))) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 >= X1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = L >= L = mark#(s(length(L))) mark#(U51(X1,X2)) = X1 >= X1 = mark#(X1) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = L >= L = mark#(U62(isNat(N),L)) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 0 >= 0 = mark#(U52(isNatList(V2))) mark#(U41(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U31(X)) = X + 1 >= X + 1 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 1 >= 1 = mark#(U42(isNatIList(V2))) mark#(U31(X)) = X + 1 >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) mark#(U11(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) 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())) = 1 >= 0 = mark(tt()) active(U41(tt(),V2)) = 1 >= 1 = 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)) = L >= L = mark(U62(isNat(N),L)) active(U62(tt(),L)) = L >= L = 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)) = 1 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = 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))) = L + N >= L = mark(U61(isNatList(L),L,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = X >= X = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X)) = X + 1 >= X + 1 = active(U31(mark(X))) mark(U41(X1,X2)) = X1 + 1 >= X1 + 1 = active(U41(mark(X1),X2)) mark(U42(X)) = X >= X = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = X1 >= X1 = active(U51(mark(X1),X2)) mark(U52(X)) = X >= X = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = X1 + X2 >= X1 + X2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 + X2 >= X1 + X2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X >= X = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X)) = X >= X = U11(X) U11(active(X)) = X >= X = U11(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X)) = X + 1 >= X + 1 = U31(X) U31(active(X)) = X + 1 >= X + 1 = U31(X) U41(mark(X1),X2) = X1 + 1 >= X1 + 1 = U41(X1,X2) U41(X1,mark(X2)) = X1 + 1 >= X1 + 1 = U41(X1,X2) U41(active(X1),X2) = X1 + 1 >= X1 + 1 = U41(X1,X2) U41(X1,active(X2)) = X1 + 1 >= X1 + 1 = U41(X1,X2) U42(mark(X)) = X >= X = U42(X) U42(active(X)) = X >= X = U42(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,mark(X2)) = X1 >= X1 = U51(X1,X2) U51(active(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,active(X2)) = X1 >= X1 = U51(X1,X2) U52(mark(X)) = X >= X = U52(X) U52(active(X)) = X >= X = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = X1 + X2 >= X1 + X2 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 + X2 >= X1 + X2 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 + X2 >= X1 + X2 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 + X2 >= X1 + X2 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 + X2 >= X1 + X2 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 + X2 >= X1 + X2 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(active(X1),X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) U62(X1,active(X2)) = X1 + X2 >= X1 + X2 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X >= X = length(X) length(active(X)) = X >= X = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) mark#(length(X)) -> mark#(X) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> active#(U52(mark(X))) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](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 + x2 + 1, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(length(cons(N,L))) = L + N + 1 >= L + N + 1 = mark#(U61(isNatList(L),L,N)) mark#(length(X)) = X + 1 >= X = mark#(X) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(isNat(V1),V2)) mark#(U62(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatList(V))) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) mark#(U52(X)) = X >= X = active#(U52(mark(X))) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatList(V1))) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 >= X1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = L + 1 >= L + 1 = mark#(s(length(L))) mark#(U51(X1,X2)) = X1 >= X1 = mark#(X1) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = L + N + 1 >= L + 1 = mark#(U62(isNat(N),L)) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = 0 >= 0 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 0 >= 0 = mark#(U52(isNatList(V2))) mark#(U31(X)) = 0 >= 0 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 0 >= 0 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = X >= X = mark#(X) mark#(U11(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(zeros()) = 0 >= 0 = active#(zeros()) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) 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)) = 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())) = 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())) = 2 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + N + 1 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = X >= X = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = X >= X = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = X1 >= X1 = active(U51(mark(X1),X2)) mark(U52(X)) = X >= X = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X)) = X >= X = U11(X) U11(active(X)) = X >= X = U11(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = X >= X = U42(X) U42(active(X)) = X >= X = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,mark(X2)) = X1 >= X1 = U51(X1,X2) U51(active(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,active(X2)) = X1 >= X1 = U51(X1,X2) U52(mark(X)) = X >= X = U52(X) U52(active(X)) = X >= X = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) U62(X1,mark(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) U62(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) U62(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> active#(U52(mark(X))) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zeros()) -> active#(zeros()) active#(zeros()) -> mark#(cons(0(),zeros())) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [nil] = 0, [s](x0) = x0, [length](x0) = 0, [U62](x0, x1) = 0, [isNat](x0) = 0, [U61](x0, x1, x2) = 0, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(length(cons(N,L))) = 0 >= 0 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 0 >= 0 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatList(V))) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) mark#(U52(X)) = X >= 0 = active#(U52(mark(X))) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatList(V1))) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 >= 0 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2)) = X1 >= X1 = mark#(X1) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = 0 >= 0 = mark#(U62(isNat(N),L)) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = 0 >= 0 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 0 >= 0 = mark#(U52(isNatList(V2))) mark#(U31(X)) = X >= 0 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 0 >= 0 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = X >= X = mark#(X) mark#(U11(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 >= X1 = mark#(X1) mark#(zeros()) = 1 >= 0 = active#(zeros()) active#(zeros()) = 0 >= 0 = mark#(cons(0(),zeros())) active(zeros()) = 1 >= 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)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 >= X1 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = X >= X = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X)) = X >= X = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = X >= X = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = X1 >= X1 = active(U51(mark(X1),X2)) mark(U52(X)) = X >= X = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = X1 >= X1 = cons(X1,X2) cons(X1,mark(X2)) = X1 >= X1 = cons(X1,X2) cons(active(X1),X2) = X1 >= X1 = cons(X1,X2) cons(X1,active(X2)) = X1 >= X1 = cons(X1,X2) U11(mark(X)) = X >= X = U11(X) U11(active(X)) = X >= X = U11(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X)) = X >= X = U31(X) U31(active(X)) = X >= X = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = X >= X = U42(X) U42(active(X)) = X >= X = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,mark(X2)) = X1 >= X1 = U51(X1,X2) U51(active(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,active(X2)) = X1 >= X1 = U51(X1,X2) U52(mark(X)) = X >= X = U52(X) U52(active(X)) = X >= X = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> active#(U52(mark(X))) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) active#(zeros()) -> mark#(cons(0(),zeros())) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 0, [active#](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) = 1, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = 0, [U11](x0) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 1, [0] = 0, [active](x0) = 0, [zeros] = 1 orientation: mark#(s(X)) = 0 >= 0 = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(length(cons(N,L))) = 0 >= 0 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 0 >= 0 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatList(V))) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) mark#(U52(X)) = 0 >= 0 = active#(U52(mark(X))) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatList(V1))) mark#(U52(X)) = 0 >= 0 = mark#(X) mark#(U51(X1,X2)) = 0 >= 0 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2)) = 0 >= 0 = mark#(X1) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = 0 >= 0 = mark#(U62(isNat(N),L)) mark#(U42(X)) = 0 >= 0 = mark#(X) mark#(U41(X1,X2)) = 0 >= 0 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 0 >= 0 = mark#(U52(isNatList(V2))) mark#(U31(X)) = 0 >= 0 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 0 >= 0 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = 0 >= 0 = mark#(X) mark#(U11(X)) = 0 >= 0 = mark#(X) mark#(cons(X1,X2)) = 0 >= 0 = mark#(X1) active#(zeros()) = 1 >= 0 = mark#(cons(0(),zeros())) 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)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = 0 >= 0 = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = 0 >= 0 = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X)) = 0 >= 0 = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 1 >= 1 = U42(X) U42(active(X)) = 1 >= 1 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> active#(U52(mark(X))) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 1, [U61](x0, x1, x2) = 1, [U52](x0) = 0, [isNatList](x0) = 1, [U51](x0, x1) = 1, [U42](x0) = 0, [isNatIList](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = 1, [U21](x0) = 1, [U11](x0) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(length(cons(N,L))) = 1 >= 1 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatList(V))) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(U52(X)) = 1 >= 0 = active#(U52(mark(X))) active#(isNat(length(V1))) = 1 >= 1 = mark#(U11(isNatList(V1))) mark#(U52(X)) = 1 >= 1 = mark#(X) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2)) = 1 >= 1 = mark#(X1) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = 1 >= 1 = mark#(U62(isNat(N),L)) mark#(U42(X)) = 1 >= 1 = mark#(X) mark#(U41(X1,X2)) = 1 >= 1 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 1 >= 1 = mark#(U52(isNatList(V2))) mark#(U31(X)) = 1 >= 1 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 1 >= 1 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = 1 >= 1 = mark#(X) mark#(U11(X)) = 1 >= 1 = mark#(X) mark#(cons(X1,X2)) = 1 >= 1 = mark#(X1) 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)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = 0 >= 0 = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = 0 >= 0 = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X)) = 0 >= 0 = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 1 >= 1 = U21(X) U21(active(X)) = 1 >= 1 = U21(X) U31(mark(X)) = 1 >= 1 = U31(X) U31(active(X)) = 1 >= 1 = U31(X) U41(mark(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,mark(X2)) = 1 >= 1 = U41(X1,X2) U41(active(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,active(X2)) = 1 >= 1 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = 0, [nil] = 0, [s](x0) = x0, [length](x0) = 0, [U62](x0, x1) = 0, [isNat](x0) = 0, [U61](x0, x1, x2) = 0, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = x0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + 1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(length(cons(N,L))) = 0 >= 0 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 0 >= 0 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatList(V))) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatList(V1))) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 >= 0 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2)) = X1 >= X1 = mark#(X1) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = 0 >= 0 = mark#(U62(isNat(N),L)) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = 0 >= 0 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 0 >= 0 = mark#(U52(isNatList(V2))) mark#(U31(X)) = 0 >= 0 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 0 >= 0 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = X >= X = mark#(X) mark#(U11(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + 1 >= X1 = mark#(X1) active(zeros()) = 1 >= 1 = 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)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = X >= X = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = X >= X = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = X1 >= X1 = active(U51(mark(X1),X2)) mark(U52(X)) = X >= X = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(X1,mark(X2)) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(active(X1),X2) = X1 + 1 >= X1 + 1 = cons(X1,X2) cons(X1,active(X2)) = X1 + 1 >= X1 + 1 = cons(X1,X2) U11(mark(X)) = X >= X = U11(X) U11(active(X)) = X >= X = U11(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = X >= X = U42(X) U42(active(X)) = X >= X = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,mark(X2)) = X1 >= X1 = U51(X1,X2) U51(active(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,active(X2)) = X1 >= X1 = U51(X1,X2) U52(mark(X)) = X >= X = U52(X) U52(active(X)) = X >= X = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) mark#(U31(X)) -> active#(U31(mark(X))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 1, [active#](x0) = x0, [nil] = 0, [s](x0) = 0, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 1, [U61](x0, x1, x2) = 1, [U52](x0) = 0, [isNatList](x0) = 1, [U51](x0, x1) = 1, [U42](x0) = 0, [isNatIList](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = 0, [U21](x0) = 0, [U11](x0) = 0, [tt] = 0, [mark](x0) = 0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = 0, [zeros] = 0 orientation: mark#(s(X)) = 1 >= 1 = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(length(cons(N,L))) = 1 >= 1 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 1 >= 1 = active#(isNat(X)) active#(isNat(s(V1))) = 1 >= 1 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 1 >= 1 = mark#(U31(isNatList(V))) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) active#(isNat(length(V1))) = 1 >= 1 = mark#(U11(isNatList(V1))) mark#(U52(X)) = 1 >= 1 = mark#(X) mark#(U51(X1,X2)) = 1 >= 1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2)) = 1 >= 1 = mark#(X1) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = 1 >= 1 = mark#(U62(isNat(N),L)) mark#(U42(X)) = 1 >= 1 = mark#(X) mark#(U41(X1,X2)) = 1 >= 1 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 1 >= 1 = mark#(U52(isNatList(V2))) mark#(U31(X)) = 1 >= 0 = active#(U31(mark(X))) active#(U41(tt(),V2)) = 1 >= 1 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = 1 >= 1 = mark#(X) mark#(U11(X)) = 1 >= 1 = mark#(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)) = 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)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = 0 >= 0 = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = 0 >= 0 = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = 0 >= 0 = active(U51(mark(X1),X2)) mark(U52(X)) = 0 >= 0 = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = 0 >= 0 = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,mark(X2)) = 1 >= 1 = U41(X1,X2) U41(active(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,active(X2)) = 1 >= 1 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,mark(X2)) = 1 >= 1 = U51(X1,X2) U51(active(X1),X2) = 1 >= 1 = U51(X1,X2) U51(X1,active(X2)) = 1 >= 1 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [nil] = 0, [s](x0) = x0, [length](x0) = x0 + 1, [U62](x0, x1) = x1 + 1, [isNat](x0) = x0, [U61](x0, x1, x2) = x1 + x2 + 1, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0 + x1, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = x0 + 1, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = X + 1 >= X + 1 = active#(length(mark(X))) active#(length(cons(N,L))) = L + N + 1 >= L + N + 1 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = X2 + 1 >= X2 + 1 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 0 >= 0 = mark#(U31(isNatList(V))) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(isNat(length(V1))) = V1 + 1 >= V1 + 1 = mark#(U11(isNatList(V1))) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = L + 1 >= L + 1 = mark#(s(length(L))) mark#(U51(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = L + N + 1 >= L + 1 = mark#(U62(isNat(N),L)) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = 0 >= 0 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = V2 >= V2 = mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) = 0 >= 0 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = X >= X = mark#(X) mark#(U11(X)) = X + 1 >= X = mark#(X) active(zeros()) = 0 >= 0 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 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)) = V2 >= V2 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = 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())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = V1 + 1 >= V1 + 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= V1 = 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))) = V1 + V2 >= V1 + V2 = mark(U51(isNat(V1),V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = L + N + 1 >= L + N + 1 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = X + 1 >= X + 1 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = X >= X = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = X1 + X2 >= X1 + X2 = active(U51(mark(X1),X2)) mark(U52(X)) = X >= X = active(U52(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = X2 + 1 >= X2 + 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = X + 1 >= X + 1 = active(length(mark(X))) mark(nil()) = 0 >= 0 = active(nil()) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X)) = X + 1 >= X + 1 = U11(X) U11(active(X)) = X + 1 >= X + 1 = U11(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = X >= X = U42(X) U42(active(X)) = X >= X = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(active(X1),X2) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(X1,active(X2)) = X1 + X2 >= X1 + X2 = U51(X1,X2) U52(mark(X)) = X >= X = U52(X) U52(active(X)) = X >= X = U52(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U61(mark(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3) U62(mark(X1),X2) = X2 + 1 >= X2 + 1 = U62(X1,X2) U62(X1,mark(X2)) = X2 + 1 >= X2 + 1 = U62(X1,X2) U62(active(X1),X2) = X2 + 1 >= X2 + 1 = U62(X1,X2) U62(X1,active(X2)) = X2 + 1 >= X2 + 1 = U62(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = X + 1 >= X + 1 = length(X) length(active(X)) = X + 1 >= X + 1 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) -> mark#(U31(isNatList(V))) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = 0, [U62](x0, x1) = 0, [isNat](x0) = 0, [U61](x0, x1, x2) = 0, [U52](x0) = x0, [isNatList](x0) = 0, [U51](x0, x1) = x0, [U42](x0) = x0, [isNatIList](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(length(cons(N,L))) = 0 >= 0 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = 0 >= 0 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 0 >= 0 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 1 >= 1 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) active#(isNatIList(V)) = 1 >= 0 = mark#(U31(isNatList(V))) mark#(isNatList(X)) = 0 >= 0 = active#(isNatList(X)) active#(isNat(length(V1))) = 0 >= 0 = mark#(U11(isNatList(V1))) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 >= X1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2)) = X1 >= X1 = mark#(X1) mark#(isNatIList(X)) = 1 >= 1 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = 0 >= 0 = mark#(U62(isNat(N),L)) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = 1 >= 1 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 0 >= 0 = mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) = 1 >= 1 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = X >= X = mark#(X) active(zeros()) = 1 >= 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)) = 1 >= 1 = 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)) = 1 >= 0 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 1 >= 0 = mark(tt()) active(isNatIList(cons(V1,V2))) = 1 >= 1 = 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)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 0 >= 0 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = X >= X = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = X1 >= X1 = active(U51(mark(X1),X2)) mark(U52(X)) = X >= X = active(U52(mark(X))) mark(isNatList(X)) = 0 >= 0 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,mark(X2)) = 1 >= 1 = U41(X1,X2) U41(active(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,active(X2)) = 1 >= 1 = U41(X1,X2) U42(mark(X)) = X >= X = U42(X) U42(active(X)) = X >= X = U42(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,mark(X2)) = X1 >= X1 = U51(X1,X2) U51(active(X1),X2) = X1 >= X1 = U51(X1,X2) U51(X1,active(X2)) = X1 >= X1 = U51(X1,X2) U52(mark(X)) = X >= X = U52(X) U52(active(X)) = X >= X = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(isNatList(X)) -> active#(isNatList(X)) active#(isNat(length(V1))) -> mark#(U11(isNatList(V1))) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = x0, [U61](x0, x1, x2) = 1, [U52](x0) = x0, [isNatList](x0) = x0, [U51](x0, x1) = x0 + x1, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = x0 + x1, [0] = 0, [active](x0) = x0, [zeros] = 0 orientation: mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 1 >= 1 = active#(length(mark(X))) active#(length(cons(N,L))) = 1 >= 1 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = X >= X = active#(isNat(X)) active#(isNat(s(V1))) = V1 >= V1 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 1 >= 1 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = 1 >= 1 = active#(U61(mark(X1),X2,X3)) mark#(isNatList(X)) = X >= X = active#(isNatList(X)) active#(isNat(length(V1))) = 1 >= 0 = mark#(U11(isNatList(V1))) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 + X2 >= X1 + X2 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 1 >= 1 = mark#(s(length(L))) mark#(U51(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = 1 >= 1 = mark#(U62(isNat(N),L)) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = 0 >= 0 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = V2 >= V2 = mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) = 0 >= 0 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = X >= X = mark#(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)) = 0 >= 0 = mark(U42(isNatIList(V2))) active(U42(tt())) = 0 >= 0 = mark(tt()) active(U51(tt(),V2)) = V2 >= V2 = mark(U52(isNatList(V2))) active(U52(tt())) = 0 >= 0 = 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())) = 0 >= 0 = mark(tt()) active(isNat(length(V1))) = 1 >= 0 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = V1 >= V1 = 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())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = V1 + V2 >= V1 + V2 = mark(U51(isNat(V1),V2)) active(length(nil())) = 1 >= 0 = mark(0()) active(length(cons(N,L))) = 1 >= 1 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 0 >= 0 = active(zeros()) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = X >= X = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = X1 + X2 >= X1 + X2 = active(U51(mark(X1),X2)) mark(U52(X)) = X >= X = active(U52(mark(X))) mark(isNatList(X)) = X >= X = active(isNatList(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = X >= X = active(isNat(X)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,mark(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(active(X1),X2) = X1 + X2 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = X >= X = U42(X) U42(active(X)) = X >= X = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(X1,mark(X2)) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(active(X1),X2) = X1 + X2 >= X1 + X2 = U51(X1,X2) U51(X1,active(X2)) = X1 + X2 >= X1 + X2 = U51(X1,X2) U52(mark(X)) = X >= X = U52(X) U52(active(X)) = X >= X = U52(X) isNatList(mark(X)) = X >= X = isNatList(X) isNatList(active(X)) = X >= X = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = X >= X = isNat(X) isNat(active(X)) = X >= X = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(U51(X1,X2)) -> mark#(X1) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [active#](x0) = x0, [nil] = 1, [s](x0) = x0, [length](x0) = 0, [U62](x0, x1) = 0, [isNat](x0) = 0, [U61](x0, x1, x2) = 0, [U52](x0) = x0, [isNatList](x0) = 1, [U51](x0, x1) = x0 + 1, [U42](x0) = x0, [isNatIList](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = 0, [tt] = 0, [mark](x0) = x0, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0, [zeros] = 1 orientation: mark#(s(X)) = X >= X = mark#(X) mark#(length(X)) = 0 >= 0 = active#(length(mark(X))) active#(length(cons(N,L))) = 0 >= 0 = mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) = 1 >= 1 = mark#(U51(isNat(V1),V2)) mark#(isNat(X)) = 0 >= 0 = active#(isNat(X)) active#(isNat(s(V1))) = 0 >= 0 = mark#(U21(isNat(V1))) mark#(U62(X1,X2)) = 0 >= 0 = active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) = 0 >= 0 = mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) = 0 >= 0 = active#(U61(mark(X1),X2,X3)) mark#(isNatList(X)) = 1 >= 1 = active#(isNatList(X)) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 + 1 >= X1 + 1 = active#(U51(mark(X1),X2)) active#(U62(tt(),L)) = 0 >= 0 = mark#(s(length(L))) mark#(U51(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(isNatIList(X)) = 0 >= 0 = active#(isNatIList(X)) active#(U61(tt(),L,N)) = 0 >= 0 = mark#(U62(isNat(N),L)) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = 0 >= 0 = active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) = 1 >= 1 = mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) = 0 >= 0 = mark#(U42(isNatIList(V2))) mark#(U21(X)) = X >= X = mark#(X) active(zeros()) = 1 >= 1 = 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)) = 1 >= 1 = 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())) = 1 >= 0 = mark(tt()) active(isNatList(cons(V1,V2))) = 1 >= 1 = 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)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 0 >= 0 = active(0()) mark(U11(X)) = 0 >= 0 = active(U11(mark(X))) mark(tt()) = 0 >= 0 = active(tt()) mark(U21(X)) = X >= X = active(U21(mark(X))) mark(U31(X)) = 0 >= 0 = active(U31(mark(X))) mark(U41(X1,X2)) = 0 >= 0 = active(U41(mark(X1),X2)) mark(U42(X)) = X >= X = active(U42(mark(X))) mark(isNatIList(X)) = 0 >= 0 = active(isNatIList(X)) mark(U51(X1,X2)) = X1 + 1 >= X1 + 1 = active(U51(mark(X1),X2)) mark(U52(X)) = X >= X = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 0 >= 0 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 0 >= 0 = active(U62(mark(X1),X2)) mark(isNat(X)) = 0 >= 0 = active(isNat(X)) mark(s(X)) = X >= X = active(s(mark(X))) mark(length(X)) = 0 >= 0 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = X >= X = U21(X) U21(active(X)) = X >= X = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = X >= X = U42(X) U42(active(X)) = X >= X = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = X1 + 1 >= X1 + 1 = U51(X1,X2) U51(X1,mark(X2)) = X1 + 1 >= X1 + 1 = U51(X1,X2) U51(active(X1),X2) = X1 + 1 >= X1 + 1 = U51(X1,X2) U51(X1,active(X2)) = X1 + 1 >= X1 + 1 = U51(X1,X2) U52(mark(X)) = X >= X = U52(X) U52(active(X)) = X >= X = U52(X) isNatList(mark(X)) = 1 >= 1 = isNatList(X) isNatList(active(X)) = 1 >= 1 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = X >= X = s(X) s(active(X)) = X >= X = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) problem: DPs: mark#(s(X)) -> mark#(X) mark#(length(X)) -> active#(length(mark(X))) active#(length(cons(N,L))) -> mark#(U61(isNatList(L),L,N)) active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2)) mark#(isNat(X)) -> active#(isNat(X)) active#(isNat(s(V1))) -> mark#(U21(isNat(V1))) mark#(U62(X1,X2)) -> active#(U62(mark(X1),X2)) active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2)) mark#(U61(X1,X2,X3)) -> active#(U61(mark(X1),X2,X3)) mark#(isNatList(X)) -> active#(isNatList(X)) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2)) active#(U62(tt(),L)) -> mark#(s(length(L))) mark#(isNatIList(X)) -> active#(isNatIList(X)) active#(U61(tt(),L,N)) -> mark#(U62(isNat(N),L)) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2)) active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2))) active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2))) mark#(U21(X)) -> mark#(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Open DPs: cons#(active(X1),X2) -> cons#(X1,X2) cons#(X1,active(X2)) -> cons#(X1,X2) cons#(X1,mark(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [cons#](x0, x1) = x0 + x1 + 1, [nil] = 1, [s](x0) = 0, [length](x0) = 1, [U62](x0, x1) = 0, [isNat](x0) = 1, [U61](x0, x1, x2) = 0, [U52](x0) = 0, [isNatList](x0) = 0, [U51](x0, x1) = 0, [U42](x0) = 1, [isNatIList](x0) = x0 + 1, [U41](x0, x1) = 1, [U31](x0) = 1, [U21](x0) = 0, [U11](x0) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: cons#(active(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = cons#(X1,X2) cons#(X1,active(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = cons#(X1,X2) cons#(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = cons#(X1,X2) cons#(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = cons#(X1,X2) active(zeros()) = 2 >= 2 = mark(cons(0(),zeros())) active(U11(tt())) = 1 >= 1 = mark(tt()) active(U21(tt())) = 1 >= 1 = mark(tt()) active(U31(tt())) = 2 >= 1 = mark(tt()) active(U41(tt(),V2)) = 2 >= 2 = mark(U42(isNatIList(V2))) active(U42(tt())) = 2 >= 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())) = 2 >= 1 = mark(tt()) active(isNat(length(V1))) = 2 >= 1 = mark(U11(isNatList(V1))) active(isNat(s(V1))) = 2 >= 1 = mark(U21(isNat(V1))) active(isNatIList(V)) = V + 2 >= 2 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 3 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 3 >= 2 = 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 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 2 >= 2 = active(U31(mark(X))) mark(U41(X1,X2)) = 2 >= 2 = active(U41(mark(X1),X2)) mark(U42(X)) = 2 >= 2 = active(U42(mark(X))) mark(isNatIList(X)) = X + 2 >= X + 2 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = 2 >= 2 = active(isNat(X)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 2 >= 2 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 1 >= 1 = U31(X) U31(active(X)) = 1 >= 1 = U31(X) U41(mark(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,mark(X2)) = 1 >= 1 = U41(X1,X2) U41(active(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,active(X2)) = 1 >= 1 = U41(X1,X2) U42(mark(X)) = 1 >= 1 = U42(X) U42(active(X)) = 1 >= 1 = U42(X) isNatIList(mark(X)) = X + 2 >= X + 1 = isNatIList(X) isNatIList(active(X)) = X + 2 >= X + 1 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 1 >= 1 = isNat(X) isNat(active(X)) = 1 >= 1 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: isNatIList#(mark(X)) -> isNatIList#(X) isNatIList#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatIList#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: isNatIList#(mark(X)) = X + 2 >= X + 1 = isNatIList#(X) isNatIList#(active(X)) = X + 2 >= X + 1 = isNatIList#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U42#(mark(X)) -> U42#(X) U42#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U42#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U42#(mark(X)) = X + 2 >= X + 1 = U42#(X) U42#(active(X)) = X + 2 >= X + 1 = U42#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: isNatList#(mark(X)) -> isNatList#(X) isNatList#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [isNatList#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: isNatList#(mark(X)) = X + 2 >= X + 1 = isNatList#(X) isNatList#(active(X)) = X + 2 >= X + 1 = isNatList#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U52#(mark(X)) -> U52#(X) U52#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U52#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U52#(mark(X)) = X + 2 >= X + 1 = U52#(X) U52#(active(X)) = X + 2 >= X + 1 = U52#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: isNat#(mark(X)) -> isNat#(X) isNat#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: isNat#(mark(X)) = X + 2 >= X + 1 = isNat#(X) isNat#(active(X)) = X + 2 >= X + 1 = isNat#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U62#(mark(X1),X2) -> U62#(X1,X2) U62#(X1,active(X2)) -> U62#(X1,X2) U62#(active(X1),X2) -> U62#(X1,X2) U62#(X1,mark(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U62#](x0, x1) = x0 + x1 + 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U62#(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = U62#(X1,X2) U62#(X1,active(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = U62#(X1,X2) U62#(active(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = U62#(X1,X2) U62#(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = U62#(X1,X2) active(zeros()) = 2 >= 2 = 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)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: length#(mark(X)) -> length#(X) length#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [length#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: length#(mark(X)) = X + 2 >= X + 1 = length#(X) length#(active(X)) = X + 2 >= X + 1 = length#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: s#(mark(X)) -> s#(X) s#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [s#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: s#(mark(X)) = X + 2 >= X + 1 = s#(X) s#(active(X)) = X + 2 >= X + 1 = s#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U11#(mark(X)) -> U11#(X) U11#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U11#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U11#(mark(X)) = X + 2 >= X + 1 = U11#(X) U11#(active(X)) = X + 2 >= X + 1 = U11#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U21#(mark(X)) -> U21#(X) U21#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U21#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U21#(mark(X)) = X + 2 >= X + 1 = U21#(X) U21#(active(X)) = X + 2 >= X + 1 = U21#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U31#(mark(X)) -> U31#(X) U31#(active(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U31#](x0) = x0 + 1, [nil] = 0, [s](x0) = 1, [length](x0) = 1, [U62](x0, x1) = 1, [isNat](x0) = 0, [U61](x0, x1, x2) = 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U31#(mark(X)) = X + 2 >= X + 1 = U31#(X) U31#(active(X)) = X + 2 >= X + 1 = U31#(X) active(zeros()) = 2 >= 2 = 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)) = 2 >= 2 = mark(U62(isNat(N),L)) active(U62(tt(),L)) = 2 >= 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())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 2 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 2 >= 2 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 2 >= 2 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 2 >= 2 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 1 >= 1 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 1 >= 1 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 1 >= 1 = U61(X1,X2,X3) U62(mark(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,mark(X2)) = 1 >= 1 = U62(X1,X2) U62(active(X1),X2) = 1 >= 1 = U62(X1,X2) U62(X1,active(X2)) = 1 >= 1 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 1 >= 1 = s(X) s(active(X)) = 1 >= 1 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U41#(mark(X1),X2) -> U41#(X1,X2) U41#(X1,active(X2)) -> U41#(X1,X2) U41#(active(X1),X2) -> U41#(X1,X2) U41#(X1,mark(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U41#](x0, x1) = x0 + x1 + 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U41#(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = U41#(X1,X2) U41#(X1,active(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = U41#(X1,X2) U41#(active(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = U41#(X1,X2) U41#(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = U41#(X1,X2) active(zeros()) = 2 >= 2 = 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)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(active(X1),X2) -> U51#(X1,X2) U51#(X1,mark(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U51#](x0, x1) = x0 + x1 + 1, [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) = x0 + 1, [cons](x0, x1) = 1, [0] = 0, [active](x0) = x0 + 1, [zeros] = 1 orientation: U51#(mark(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = U51#(X1,X2) U51#(X1,active(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = U51#(X1,X2) U51#(active(X1),X2) = X1 + X2 + 2 >= X1 + X2 + 1 = U51#(X1,X2) U51#(X1,mark(X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = U51#(X1,X2) active(zeros()) = 2 >= 2 = 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)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 2 >= 2 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(nil()) = 1 >= 1 = active(nil()) cons(mark(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,mark(X2)) = 1 >= 1 = cons(X1,X2) cons(active(X1),X2) = 1 >= 1 = cons(X1,X2) cons(X1,active(X2)) = 1 >= 1 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed DPs: U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U61#(X1,mark(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1, x2) = x1, [nil] = 1, [s](x0) = 0, [length](x0) = 1, [U62](x0, x1) = 0, [isNat](x0) = 0, [U61](x0, x1, x2) = 0, [U52](x0) = 0, [isNatList](x0) = x0 + 1, [U51](x0, x1) = 0, [U42](x0) = 0, [isNatIList](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = 0, [U21](x0) = 0, [U11](x0) = 0, [tt] = 0, [mark](x0) = x0 + 1, [cons](x0, x1) = 0, [0] = 1, [active](x0) = x0 + 1, [zeros] = 1 orientation: U61#(mark(X1),X2,X3) = X2 >= X2 = U61#(X1,X2,X3) U61#(X1,X2,active(X3)) = X2 >= X2 = U61#(X1,X2,X3) U61#(X1,active(X2),X3) = X2 + 1 >= X2 = U61#(X1,X2,X3) U61#(active(X1),X2,X3) = X2 >= X2 = U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) = X2 >= X2 = U61#(X1,X2,X3) U61#(X1,mark(X2),X3) = X2 + 1 >= X2 = U61#(X1,X2,X3) active(zeros()) = 2 >= 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)) = 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())) = 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)) = 2 >= 1 = mark(U31(isNatList(V))) active(isNatIList(zeros())) = 2 >= 1 = mark(tt()) active(isNatIList(cons(V1,V2))) = 2 >= 2 = mark(U41(isNat(V1),V2)) active(isNatList(nil())) = 3 >= 1 = mark(tt()) active(isNatList(cons(V1,V2))) = 2 >= 1 = mark(U51(isNat(V1),V2)) active(length(nil())) = 2 >= 2 = mark(0()) active(length(cons(N,L))) = 2 >= 1 = mark(U61(isNatList(L),L,N)) mark(zeros()) = 2 >= 2 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 2 >= 2 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 2 >= 2 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 2 >= 2 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = X + 2 >= X + 2 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 2 >= 2 = active(length(mark(X))) mark(nil()) = 2 >= 2 = active(nil()) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,mark(X2)) = 1 >= 1 = U41(X1,X2) U41(active(X1),X2) = 1 >= 1 = U41(X1,X2) U41(X1,active(X2)) = 1 >= 1 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 1 >= 1 = isNatIList(X) isNatIList(active(X)) = 1 >= 1 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = X + 2 >= X + 1 = isNatList(X) isNatList(active(X)) = X + 2 >= X + 1 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 1 >= 1 = length(X) length(active(X)) = 1 >= 1 = length(X) problem: DPs: U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,X2,mark(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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [U61#](x0, x1, x2) = x0 + x2 + 1, [nil] = 1, [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) = x0 + 1, [cons](x0, x1) = 0, [0] = 0, [active](x0) = x0 + 1, [zeros] = 0 orientation: U61#(mark(X1),X2,X3) = X1 + X3 + 2 >= X1 + X3 + 1 = U61#(X1,X2,X3) U61#(X1,X2,active(X3)) = X1 + X3 + 2 >= X1 + X3 + 1 = U61#(X1,X2,X3) U61#(active(X1),X2,X3) = X1 + X3 + 2 >= X1 + X3 + 1 = U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) = X1 + X3 + 2 >= X1 + X3 + 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)) = 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)) mark(zeros()) = 1 >= 1 = active(zeros()) mark(cons(X1,X2)) = 1 >= 1 = active(cons(mark(X1),X2)) mark(0()) = 1 >= 1 = active(0()) mark(U11(X)) = 1 >= 1 = active(U11(mark(X))) mark(tt()) = 1 >= 1 = active(tt()) mark(U21(X)) = 1 >= 1 = active(U21(mark(X))) mark(U31(X)) = 1 >= 1 = active(U31(mark(X))) mark(U41(X1,X2)) = 1 >= 1 = active(U41(mark(X1),X2)) mark(U42(X)) = 1 >= 1 = active(U42(mark(X))) mark(isNatIList(X)) = 1 >= 1 = active(isNatIList(X)) mark(U51(X1,X2)) = 1 >= 1 = active(U51(mark(X1),X2)) mark(U52(X)) = 1 >= 1 = active(U52(mark(X))) mark(isNatList(X)) = 1 >= 1 = active(isNatList(X)) mark(U61(X1,X2,X3)) = 1 >= 1 = active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) = 1 >= 1 = active(U62(mark(X1),X2)) mark(isNat(X)) = 1 >= 1 = active(isNat(X)) mark(s(X)) = 1 >= 1 = active(s(mark(X))) mark(length(X)) = 1 >= 1 = active(length(mark(X))) mark(nil()) = 2 >= 2 = active(nil()) cons(mark(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,mark(X2)) = 0 >= 0 = cons(X1,X2) cons(active(X1),X2) = 0 >= 0 = cons(X1,X2) cons(X1,active(X2)) = 0 >= 0 = cons(X1,X2) U11(mark(X)) = 0 >= 0 = U11(X) U11(active(X)) = 0 >= 0 = U11(X) U21(mark(X)) = 0 >= 0 = U21(X) U21(active(X)) = 0 >= 0 = U21(X) U31(mark(X)) = 0 >= 0 = U31(X) U31(active(X)) = 0 >= 0 = U31(X) U41(mark(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,mark(X2)) = 0 >= 0 = U41(X1,X2) U41(active(X1),X2) = 0 >= 0 = U41(X1,X2) U41(X1,active(X2)) = 0 >= 0 = U41(X1,X2) U42(mark(X)) = 0 >= 0 = U42(X) U42(active(X)) = 0 >= 0 = U42(X) isNatIList(mark(X)) = 0 >= 0 = isNatIList(X) isNatIList(active(X)) = 0 >= 0 = isNatIList(X) U51(mark(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,mark(X2)) = 0 >= 0 = U51(X1,X2) U51(active(X1),X2) = 0 >= 0 = U51(X1,X2) U51(X1,active(X2)) = 0 >= 0 = U51(X1,X2) U52(mark(X)) = 0 >= 0 = U52(X) U52(active(X)) = 0 >= 0 = U52(X) isNatList(mark(X)) = 0 >= 0 = isNatList(X) isNatList(active(X)) = 0 >= 0 = isNatList(X) U61(mark(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,mark(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,mark(X3)) = 0 >= 0 = U61(X1,X2,X3) U61(active(X1),X2,X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,active(X2),X3) = 0 >= 0 = U61(X1,X2,X3) U61(X1,X2,active(X3)) = 0 >= 0 = U61(X1,X2,X3) U62(mark(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,mark(X2)) = 0 >= 0 = U62(X1,X2) U62(active(X1),X2) = 0 >= 0 = U62(X1,X2) U62(X1,active(X2)) = 0 >= 0 = U62(X1,X2) isNat(mark(X)) = 0 >= 0 = isNat(X) isNat(active(X)) = 0 >= 0 = isNat(X) s(mark(X)) = 0 >= 0 = s(X) s(active(X)) = 0 >= 0 = s(X) length(mark(X)) = 0 >= 0 = length(X) length(active(X)) = 0 >= 0 = length(X) 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)) mark(zeros()) -> active(zeros()) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(0()) -> active(0()) mark(U11(X)) -> active(U11(mark(X))) mark(tt()) -> active(tt()) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1,X2)) -> active(U41(mark(X1),X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2)) -> active(U62(mark(X1),X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(nil()) -> active(nil()) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1),X2) -> U41(X1,X2) U41(X1,mark(X2)) -> U41(X1,X2) U41(active(X1),X2) -> U41(X1,X2) U41(X1,active(X2)) -> U41(X1,X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U51(active(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U62(mark(X1),X2) -> U62(X1,X2) U62(X1,mark(X2)) -> U62(X1,X2) U62(active(X1),X2) -> U62(X1,X2) U62(X1,active(X2)) -> U62(X1,X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) Qed