MAYBE Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Proof: DP Processor: DPs: a__U41#(tt(),V2) -> a__isNatIList#(V2) a__U41#(tt(),V2) -> a__U42#(a__isNatIList(V2)) a__U51#(tt(),V2) -> a__isNatList#(V2) a__U51#(tt(),V2) -> a__U52#(a__isNatList(V2)) a__U61#(tt(),L,N) -> a__isNat#(N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> mark#(L) a__U62#(tt(),L) -> a__length#(mark(L)) a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatList(V1)) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__isNatIList#(V) -> a__isNatList#(V) a__isNatIList#(V) -> a__U31#(a__isNatList(V)) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) mark#(zeros()) -> a__zeros#() mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> a__U11#(mark(X)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> a__U21#(mark(X)) mark#(U31(X)) -> mark#(X) mark#(U31(X)) -> a__U31#(mark(X)) mark#(U41(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U42(X)) -> mark#(X) mark#(U42(X)) -> a__U42#(mark(X)) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X)) -> mark#(X) mark#(U52(X)) -> a__U52#(mark(X)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) EDG Processor: DPs: a__U41#(tt(),V2) -> a__isNatIList#(V2) a__U41#(tt(),V2) -> a__U42#(a__isNatIList(V2)) a__U51#(tt(),V2) -> a__isNatList#(V2) a__U51#(tt(),V2) -> a__U52#(a__isNatList(V2)) a__U61#(tt(),L,N) -> a__isNat#(N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> mark#(L) a__U62#(tt(),L) -> a__length#(mark(L)) a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNat#(length(V1)) -> a__U11#(a__isNatList(V1)) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__isNatIList#(V) -> a__isNatList#(V) a__isNatIList#(V) -> a__U31#(a__isNatList(V)) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) mark#(zeros()) -> a__zeros#() mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> a__U11#(mark(X)) mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> a__U21#(mark(X)) mark#(U31(X)) -> mark#(X) mark#(U31(X)) -> a__U31#(mark(X)) mark#(U41(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U42(X)) -> mark#(X) mark#(U42(X)) -> a__U42#(mark(X)) mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X)) -> mark#(X) mark#(U52(X)) -> a__U52#(mark(X)) mark#(isNatList(X)) -> a__isNatList#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) graph: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) -> a__U61#(tt(),L,N) -> a__isNat#(N) a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) -> a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__length#(cons(N,L)) -> a__isNatList#(L) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(length(V1)) -> a__isNatList#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(length(V1)) -> a__U11#(a__isNatList(V1)) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__isNat#(V1) mark#(isNat(X)) -> a__isNat#(X) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> mark#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U42(X)) -> mark#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U52(X)) -> mark#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U62(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) -> a__U62#(tt(),L) -> mark#(L) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) -> a__U62#(tt(),L) -> a__length#(mark(L)) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U42(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U52(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) -> a__U61#(tt(),L,N) -> a__isNat#(N) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) -> a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) mark#(isNatList(X)) -> a__isNatList#(X) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) mark#(U52(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U52(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U52(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U52(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U52(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U52(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(U52(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U52(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U52(X)) -> mark#(X) -> mark#(U42(X)) -> mark#(X) mark#(U52(X)) -> mark#(X) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U52(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U52(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U52(X)) -> mark#(X) -> mark#(U52(X)) -> mark#(X) mark#(U52(X)) -> mark#(X) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U52(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U52(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U52(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U52(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U52(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U52(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U52(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U52(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U42(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U51(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) -> a__U51#(tt(),V2) -> a__isNatList#(V2) mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) -> a__U51#(tt(),V2) -> a__U52#(a__isNatList(V2)) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(V) -> a__isNatList#(V) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(V) -> a__U31#(a__isNatList(V)) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) mark#(isNatIList(X)) -> a__isNatIList#(X) -> a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2) mark#(U42(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U42(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U42(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U42(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U42(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U42(X)) -> mark#(X) -> mark#(U42(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U42(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U42(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U42(X)) -> mark#(X) -> mark#(U52(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U42(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U42(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U42(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U42(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U42(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U42(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U42(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U52(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U41(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) -> a__U41#(tt(),V2) -> a__isNatIList#(V2) mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) -> a__U41#(tt(),V2) -> a__U42#(a__isNatIList(V2)) mark#(U31(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U31(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U31(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U31(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U31(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U31(X)) -> mark#(X) -> mark#(U42(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U31(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U31(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U31(X)) -> mark#(X) -> mark#(U52(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U31(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U31(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U31(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U31(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U31(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U31(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U21(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U21(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U21(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U21(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U21(X)) -> mark#(X) -> mark#(U42(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U21(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U21(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U21(X)) -> mark#(X) -> mark#(U52(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U21(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U21(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U21(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U21(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U21(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U21(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U21(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(U11(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(U11(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(U11(X)) -> mark#(X) -> mark#(U42(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(U11(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(U11(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(U11(X)) -> mark#(X) -> mark#(U52(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(U11(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U11(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U11(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(U11(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U11(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(U11(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(U11(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__isNatList#(L) mark#(length(X)) -> a__length#(mark(X)) -> a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) mark#(length(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(length(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U42(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(length(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(U52(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(length(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(length(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(length(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(length(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(length(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(zeros()) -> a__zeros#() mark#(s(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U21(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U31(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U42(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(s(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(U52(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(s(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(isNat(X)) -> a__isNat#(X) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(length(X)) -> a__length#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zeros()) -> a__zeros#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U21(X)) -> a__U21#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U31(X)) -> a__U31#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U42(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U42(X)) -> a__U42#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatIList(X)) -> a__isNatIList#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U52(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U52(X)) -> a__U52#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNatList(X)) -> a__isNatList#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U62(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(isNat(X)) -> a__isNat#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(length(X)) -> a__length#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) a__U62#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__isNatList#(L) a__U62#(tt(),L) -> a__length#(mark(L)) -> a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U62#(tt(),L) -> mark#(L) -> mark#(zeros()) -> a__zeros#() a__U62#(tt(),L) -> mark#(L) -> mark#(U11(X)) -> mark#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(U11(X)) -> a__U11#(mark(X)) a__U62#(tt(),L) -> mark#(L) -> mark#(U21(X)) -> mark#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(U21(X)) -> a__U21#(mark(X)) a__U62#(tt(),L) -> mark#(L) -> mark#(U31(X)) -> mark#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(U31(X)) -> a__U31#(mark(X)) a__U62#(tt(),L) -> mark#(L) -> mark#(U41(X1,X2)) -> mark#(X1) a__U62#(tt(),L) -> mark#(L) -> mark#(U41(X1,X2)) -> a__U41#(mark(X1),X2) a__U62#(tt(),L) -> mark#(L) -> mark#(U42(X)) -> mark#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(U42(X)) -> a__U42#(mark(X)) a__U62#(tt(),L) -> mark#(L) -> mark#(isNatIList(X)) -> a__isNatIList#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(U51(X1,X2)) -> mark#(X1) a__U62#(tt(),L) -> mark#(L) -> mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2) a__U62#(tt(),L) -> mark#(L) -> mark#(U52(X)) -> mark#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(U52(X)) -> a__U52#(mark(X)) a__U62#(tt(),L) -> mark#(L) -> mark#(isNatList(X)) -> a__isNatList#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(U61(X1,X2,X3)) -> mark#(X1) a__U62#(tt(),L) -> mark#(L) -> mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) a__U62#(tt(),L) -> mark#(L) -> mark#(U62(X1,X2)) -> mark#(X1) a__U62#(tt(),L) -> mark#(L) -> mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) a__U62#(tt(),L) -> mark#(L) -> mark#(isNat(X)) -> a__isNat#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(length(X)) -> mark#(X) a__U62#(tt(),L) -> mark#(L) -> mark#(length(X)) -> a__length#(mark(X)) a__U62#(tt(),L) -> mark#(L) -> mark#(cons(X1,X2)) -> mark#(X1) a__U62#(tt(),L) -> mark#(L) -> mark#(s(X)) -> mark#(X) a__isNat#(length(V1)) -> a__isNatList#(V1) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatList#(V1) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__U11#(a__isNatList(V1)) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) -> a__U62#(tt(),L) -> mark#(L) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) -> a__U62#(tt(),L) -> a__length#(mark(L)) a__U61#(tt(),L,N) -> a__isNat#(N) -> a__isNat#(length(V1)) -> a__isNatList#(V1) a__U61#(tt(),L,N) -> a__isNat#(N) -> a__isNat#(length(V1)) -> a__U11#(a__isNatList(V1)) a__U61#(tt(),L,N) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__U61#(tt(),L,N) -> a__isNat#(N) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__U11#(a__isNatList(V1)) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) -> a__U51#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) -> a__U51#(tt(),V2) -> a__U52#(a__isNatList(V2)) a__U51#(tt(),V2) -> a__isNatList#(V2) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__U51#(tt(),V2) -> a__isNatList#(V2) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(length(V1)) -> a__U11#(a__isNatList(V1)) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__isNat#(V1) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) -> a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2) -> a__U41#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2) -> a__U41#(tt(),V2) -> a__U42#(a__isNatIList(V2)) a__isNatIList#(V) -> a__isNatList#(V) -> a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNatIList#(V) -> a__isNatList#(V) -> a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) a__U41#(tt(),V2) -> a__isNatIList#(V2) -> a__isNatIList#(V) -> a__isNatList#(V) a__U41#(tt(),V2) -> a__isNatIList#(V2) -> a__isNatIList#(V) -> a__U31#(a__isNatList(V)) a__U41#(tt(),V2) -> a__isNatIList#(V2) -> a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) a__U41#(tt(),V2) -> a__isNatIList#(V2) -> a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2) SCC Processor: #sccs: 3 #rules: 26 #arcs: 396/2116 DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) a__U62#(tt(),L) -> mark#(L) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) mark#(length(X)) -> mark#(X) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) mark#(U31(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = x0, [mark#](x0) = x0, [a__U62#](x0, x1) = x1, [a__U61#](x0, x1, x2) = x1, [isNat](x0) = 0, [U62](x0, x1) = x0 + x1, [U61](x0, x1, x2) = x0 + x1, [isNatList](x0) = 0, [U52](x0) = x0, [U51](x0, x1) = x0, [isNatIList](x0) = 1, [U42](x0) = x0, [U41](x0, x1) = x0 + 1, [U31](x0) = x0, [U21](x0) = x0, [U11](x0) = x0, [nil] = 1, [length](x0) = x0, [s](x0) = x0, [a__length](x0) = x0, [mark](x0) = x0, [a__U62](x0, x1) = x0 + x1, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = x0 + x1, [a__U52](x0) = x0, [a__isNatList](x0) = 0, [a__U51](x0, x1) = x0, [a__U42](x0) = x0, [a__isNatIList](x0) = 1, [a__U41](x0, x1) = x0 + 1, [a__U31](x0) = x0, [a__U21](x0) = x0, [a__U11](x0) = x0, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__length#(cons(N,L)) = L + N >= L = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = L >= L = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = L >= L = a__length#(mark(L)) a__U62#(tt(),L) = L >= L = mark#(L) mark#(s(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(length(X)) = X >= X = a__length#(mark(X)) mark#(length(X)) = X >= X = mark#(X) mark#(U62(X1,X2)) = X1 + X2 >= X2 = a__U62#(mark(X1),X2) mark#(U62(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 >= X2 = a__U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U31(X)) = X >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) mark#(U11(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = 1 >= 1 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = 0 >= 0 = a__U52(a__isNatList(V2)) a__U52(tt()) = 0 >= 0 = tt() a__U61(tt(),L,N) = L >= L = a__U62(a__isNat(N),L) a__U62(tt(),L) = L >= L = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = 1 >= 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__isNat(V1),V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = L + N >= L = a__U61(a__isNatList(L),L,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X)) = X >= X = a__U11(mark(X)) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X)) = X >= X = a__U31(mark(X)) mark(U41(X1,X2)) = X1 + 1 >= X1 + 1 = a__U41(mark(X1),X2) mark(U42(X)) = X >= X = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2)) = X1 >= X1 = a__U51(mark(X1),X2) mark(U52(X)) = X >= X = a__U52(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U61(X1,X2,X3)) = X1 + X2 >= X1 + X2 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 + X2 >= X1 + X2 = a__U62(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = X >= X = a__length(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X) = X >= X = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = X >= X = U31(X) a__U41(X1,X2) = X1 + 1 >= X1 + 1 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2) = X1 >= X1 = U51(X1,X2) a__U52(X) = X >= X = U52(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U61(X1,X2,X3) = X1 + X2 >= X1 + X2 = U61(X1,X2,X3) a__U62(X1,X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = X >= X = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) a__U62#(tt(),L) -> mark#(L) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) mark#(length(X)) -> mark#(X) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = x0, [mark#](x0) = x0, [a__U62#](x0, x1) = x1, [a__U61#](x0, x1, x2) = x1, [isNat](x0) = 0, [U62](x0, x1) = x0 + x1, [U61](x0, x1, x2) = x0 + x1, [isNatList](x0) = 0, [U52](x0) = x0, [U51](x0, x1) = x0, [isNatIList](x0) = 1, [U42](x0) = x0, [U41](x0, x1) = 1, [U31](x0) = x0 + 1, [U21](x0) = x0, [U11](x0) = x0, [nil] = 0, [length](x0) = x0, [s](x0) = x0, [a__length](x0) = x0, [mark](x0) = x0, [a__U62](x0, x1) = x0 + x1, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = x0 + x1, [a__U52](x0) = x0, [a__isNatList](x0) = 0, [a__U51](x0, x1) = x0, [a__U42](x0) = x0, [a__isNatIList](x0) = 1, [a__U41](x0, x1) = 1, [a__U31](x0) = x0 + 1, [a__U21](x0) = x0, [a__U11](x0) = x0, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__length#(cons(N,L)) = L + N >= L = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = L >= L = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = L >= L = a__length#(mark(L)) a__U62#(tt(),L) = L >= L = mark#(L) mark#(s(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(length(X)) = X >= X = a__length#(mark(X)) mark#(length(X)) = X >= X = mark#(X) mark#(U62(X1,X2)) = X1 + X2 >= X2 = a__U62#(mark(X1),X2) mark#(U62(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 >= X2 = a__U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X)) = X >= X = mark#(X) mark#(U31(X)) = X + 1 >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) mark#(U11(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 1 >= 0 = tt() a__U41(tt(),V2) = 1 >= 1 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = 0 >= 0 = a__U52(a__isNatList(V2)) a__U52(tt()) = 0 >= 0 = tt() a__U61(tt(),L,N) = L >= L = a__U62(a__isNat(N),L) a__U62(tt(),L) = L >= L = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__isNat(V1),V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = L + N >= L = a__U61(a__isNatList(L),L,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X)) = X >= X = a__U11(mark(X)) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X)) = X + 1 >= X + 1 = a__U31(mark(X)) mark(U41(X1,X2)) = 1 >= 1 = a__U41(mark(X1),X2) mark(U42(X)) = X >= X = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2)) = X1 >= X1 = a__U51(mark(X1),X2) mark(U52(X)) = X >= X = a__U52(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U61(X1,X2,X3)) = X1 + X2 >= X1 + X2 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 + X2 >= X1 + X2 = a__U62(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = X >= X = a__length(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X) = X >= X = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = X + 1 >= X + 1 = U31(X) a__U41(X1,X2) = 1 >= 1 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2) = X1 >= X1 = U51(X1,X2) a__U52(X) = X >= X = U52(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U61(X1,X2,X3) = X1 + X2 >= X1 + X2 = U61(X1,X2,X3) a__U62(X1,X2) = X1 + X2 >= X1 + X2 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = X >= X = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) a__U62#(tt(),L) -> mark#(L) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(length(X)) -> a__length#(mark(X)) mark#(length(X)) -> mark#(X) mark#(U62(X1,X2)) -> a__U62#(mark(X1),X2) mark#(U62(X1,X2)) -> mark#(X1) mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = x0, [mark#](x0) = x0, [a__U62#](x0, x1) = x1, [a__U61#](x0, x1, x2) = x1 + x2, [isNat](x0) = 0, [U62](x0, x1) = x0 + x1 + 1, [U61](x0, x1, x2) = x0 + x1 + x2 + 1, [isNatList](x0) = 0, [U52](x0) = x0, [U51](x0, x1) = x0, [isNatIList](x0) = 0, [U42](x0) = x0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = x0, [nil] = 0, [length](x0) = x0 + 1, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0, [a__U62](x0, x1) = x0 + x1 + 1, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = x0 + x1 + x2 + 1, [a__U52](x0) = x0, [a__isNatList](x0) = 0, [a__U51](x0, x1) = x0, [a__U42](x0) = x0, [a__isNatIList](x0) = 0, [a__U41](x0, x1) = 0, [a__U31](x0) = 0, [a__U21](x0) = x0, [a__U11](x0) = x0, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__length#(cons(N,L)) = L + N >= L + N = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = L + N >= L = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = L >= L = a__length#(mark(L)) a__U62#(tt(),L) = L >= L = mark#(L) mark#(s(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(length(X)) = X + 1 >= X = a__length#(mark(X)) mark#(length(X)) = X + 1 >= X = mark#(X) mark#(U62(X1,X2)) = X1 + X2 + 1 >= X2 = a__U62#(mark(X1),X2) mark#(U62(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X2 + X3 = a__U61#(mark(X1),X2,X3) mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 >= X1 = mark#(X1) mark#(U42(X)) = X >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) mark#(U11(X)) = X >= X = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = 0 >= 0 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = 0 >= 0 = a__U52(a__isNatList(V2)) a__U52(tt()) = 0 >= 0 = tt() a__U61(tt(),L,N) = L + N + 1 >= L + 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__isNat(V1),V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + N + 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X)) = X >= X = a__U11(mark(X)) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X)) = 0 >= 0 = a__U31(mark(X)) mark(U41(X1,X2)) = 0 >= 0 = a__U41(mark(X1),X2) mark(U42(X)) = X >= X = a__U42(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2)) = X1 >= X1 = a__U51(mark(X1),X2) mark(U52(X)) = X >= X = a__U52(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X) = X >= X = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = 0 >= 0 = U31(X) a__U41(X1,X2) = 0 >= 0 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2) = X1 >= X1 = U51(X1,X2) a__U52(X) = X >= X = U52(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U61(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = X + 1 >= X + 1 = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) a__U62#(tt(),L) -> mark#(L) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = x0 + 1, [mark#](x0) = 0, [a__U62#](x0, x1) = x0 + 1, [a__U61#](x0, x1, x2) = x0 + 1, [isNat](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [isNatList](x0) = 1, [U52](x0) = 1, [U51](x0, x1) = 1, [isNatIList](x0) = 1, [U42](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = x0, [U21](x0) = 1, [U11](x0) = 1, [nil] = 1, [length](x0) = 1, [s](x0) = 1, [a__length](x0) = 1, [mark](x0) = 1, [a__U62](x0, x1) = 1, [a__isNat](x0) = 1, [a__U61](x0, x1, x2) = 1, [a__U52](x0) = 1, [a__isNatList](x0) = 1, [a__U51](x0, x1) = 1, [a__U42](x0) = 1, [a__isNatIList](x0) = 1, [a__U41](x0, x1) = 1, [a__U31](x0) = x0, [a__U21](x0) = 1, [a__U11](x0) = 1, [tt] = 1, [cons](x0, x1) = 1, [zeros] = 1, [0] = 1, [a__zeros] = 1 orientation: a__length#(cons(N,L)) = 2 >= 2 = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = 2 >= 2 = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = 2 >= 2 = a__length#(mark(L)) a__U62#(tt(),L) = 2 >= 0 = mark#(L) mark#(s(X)) = 0 >= 0 = mark#(X) mark#(cons(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U52(X)) = 0 >= 0 = mark#(X) mark#(U51(X1,X2)) = 0 >= 0 = mark#(X1) mark#(U42(X)) = 0 >= 0 = mark#(X) mark#(U21(X)) = 0 >= 0 = mark#(X) mark#(U11(X)) = 0 >= 0 = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 1 >= 1 = tt() a__U21(tt()) = 1 >= 1 = tt() a__U31(tt()) = 1 >= 1 = tt() a__U41(tt(),V2) = 1 >= 1 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 1 >= 1 = tt() a__U51(tt(),V2) = 1 >= 1 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 1 = tt() a__U61(tt(),L,N) = 1 >= 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 1 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__isNat(V1),V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 1 >= 1 = a__U11(mark(X)) mark(U21(X)) = 1 >= 1 = a__U21(mark(X)) mark(U31(X)) = 1 >= 1 = a__U31(mark(X)) mark(U41(X1,X2)) = 1 >= 1 = a__U41(mark(X1),X2) mark(U42(X)) = 1 >= 1 = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2) mark(U52(X)) = 1 >= 1 = a__U52(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(cons(X1,X2)) = 1 >= 1 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 1 >= 1 = tt() mark(s(X)) = 1 >= 1 = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X) = 1 >= 1 = U11(X) a__U21(X) = 1 >= 1 = U21(X) a__U31(X) = X >= X = U31(X) a__U41(X1,X2) = 1 >= 1 = U41(X1,X2) a__U42(X) = 1 >= 1 = U42(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2) = 1 >= 1 = U51(X1,X2) a__U52(X) = 1 >= 1 = U52(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U61(X1,X2,X3) = 1 >= 1 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 1 = U62(X1,X2) a__isNat(X) = 1 >= 1 = isNat(X) a__length(X) = 1 >= 1 = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = 0, [mark#](x0) = x0 + 1, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [isNat](x0) = x0, [U62](x0, x1) = x1 + 1, [U61](x0, x1, x2) = x1 + 1, [isNatList](x0) = x0, [U52](x0) = x0, [U51](x0, x1) = x0 + x1, [isNatIList](x0) = x0, [U42](x0) = x0, [U41](x0, x1) = x1, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = x0 + 1, [nil] = 0, [length](x0) = x0 + 1, [s](x0) = x0, [a__length](x0) = x0 + 1, [mark](x0) = x0, [a__U62](x0, x1) = x1 + 1, [a__isNat](x0) = x0, [a__U61](x0, x1, x2) = x1 + 1, [a__U52](x0) = x0, [a__isNatList](x0) = x0, [a__U51](x0, x1) = x0 + x1, [a__U42](x0) = x0, [a__isNatIList](x0) = x0, [a__U41](x0, x1) = x1, [a__U31](x0) = 0, [a__U21](x0) = x0, [a__U11](x0) = x0 + 1, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 0, [0] = 0, [a__zeros] = 0 orientation: a__length#(cons(N,L)) = 0 >= 0 = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = 0 >= 0 = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(s(X)) = X + 1 >= X + 1 = mark#(X) mark#(cons(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(U52(X)) = X + 1 >= X + 1 = mark#(X) mark#(U51(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(U42(X)) = X + 1 >= X + 1 = mark#(X) mark#(U21(X)) = X + 1 >= X + 1 = mark#(X) mark#(U11(X)) = X + 2 >= X + 1 = mark#(X) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt()) = 1 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = V2 >= V2 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = V2 >= V2 = a__U52(a__isNatList(V2)) a__U52(tt()) = 0 >= 0 = tt() a__U61(tt(),L,N) = L + 1 >= L + 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = V1 + 1 >= V1 + 1 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = V >= 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 >= V2 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 >= V1 + V2 = a__U51(a__isNat(V1),V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X)) = X + 1 >= X + 1 = a__U11(mark(X)) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X)) = 0 >= 0 = a__U31(mark(X)) mark(U41(X1,X2)) = X2 >= X2 = a__U41(mark(X1),X2) mark(U42(X)) = X >= X = a__U42(mark(X)) mark(isNatIList(X)) = X >= X = a__isNatIList(X) mark(U51(X1,X2)) = X1 + X2 >= X1 + X2 = a__U51(mark(X1),X2) mark(U52(X)) = X >= X = a__U52(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U61(X1,X2,X3)) = X2 + 1 >= X2 + 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X2 + 1 >= X2 + 1 = a__U62(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X) = X + 1 >= X + 1 = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = 0 >= 0 = U31(X) a__U41(X1,X2) = X2 >= X2 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = X >= X = isNatIList(X) a__U51(X1,X2) = X1 + X2 >= X1 + X2 = U51(X1,X2) a__U52(X) = X >= X = U52(X) a__isNatList(X) = X >= X = isNatList(X) a__U61(X1,X2,X3) = X2 + 1 >= X2 + 1 = U61(X1,X2,X3) a__U62(X1,X2) = X2 + 1 >= X2 + 1 = U62(X1,X2) a__isNat(X) = X >= X = isNat(X) a__length(X) = X + 1 >= X + 1 = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = x0, [mark#](x0) = x0, [a__U62#](x0, x1) = x1, [a__U61#](x0, x1, x2) = x1 + x2, [isNat](x0) = 0, [U62](x0, x1) = x1, [U61](x0, x1, x2) = x1, [isNatList](x0) = 1, [U52](x0) = x0, [U51](x0, x1) = x0 + 1, [isNatIList](x0) = 0, [U42](x0) = x0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = 0, [nil] = 0, [length](x0) = x0, [s](x0) = x0, [a__length](x0) = x0, [mark](x0) = x0, [a__U62](x0, x1) = x1, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = x1, [a__U52](x0) = x0, [a__isNatList](x0) = 1, [a__U51](x0, x1) = x0 + 1, [a__U42](x0) = x0, [a__isNatIList](x0) = 0, [a__U41](x0, x1) = 0, [a__U31](x0) = 0, [a__U21](x0) = x0, [a__U11](x0) = 0, [tt] = 0, [cons](x0, x1) = x0 + x1, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__length#(cons(N,L)) = L + N >= L + N = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = L + N >= L = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = L >= L = a__length#(mark(L)) mark#(s(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U42(X)) = X >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = 0 >= 0 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = 1 >= 1 = a__U52(a__isNatList(V2)) a__U52(tt()) = 0 >= 0 = tt() a__U61(tt(),L,N) = L >= L = a__U62(a__isNat(N),L) a__U62(tt(),L) = L >= L = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__isNat(V1),V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = L + N >= L = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 0 >= 0 = a__U11(mark(X)) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X)) = 0 >= 0 = a__U31(mark(X)) mark(U41(X1,X2)) = 0 >= 0 = a__U41(mark(X1),X2) mark(U42(X)) = X >= X = a__U42(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2)) = X1 + 1 >= X1 + 1 = a__U51(mark(X1),X2) mark(U52(X)) = X >= X = a__U52(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U61(X1,X2,X3)) = X2 >= X2 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X2 >= X2 = a__U62(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = X >= X = a__length(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X) = 0 >= 0 = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = 0 >= 0 = U31(X) a__U41(X1,X2) = 0 >= 0 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2) = X1 + 1 >= X1 + 1 = U51(X1,X2) a__U52(X) = X >= X = U52(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U61(X1,X2,X3) = X2 >= X2 = U61(X1,X2,X3) a__U62(X1,X2) = X2 >= X2 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = X >= X = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = 0, [mark#](x0) = x0, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [isNat](x0) = 0, [U62](x0, x1) = 1, [U61](x0, x1, x2) = 1, [isNatList](x0) = 0, [U52](x0) = x0, [U51](x0, x1) = 0, [isNatIList](x0) = 0, [U42](x0) = x0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = 0, [nil] = 0, [length](x0) = 1, [s](x0) = x0, [a__length](x0) = 1, [mark](x0) = x0, [a__U62](x0, x1) = 1, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = 1, [a__U52](x0) = x0, [a__isNatList](x0) = 0, [a__U51](x0, x1) = 0, [a__U42](x0) = x0, [a__isNatIList](x0) = 0, [a__U41](x0, x1) = 0, [a__U31](x0) = 0, [a__U21](x0) = x0, [a__U11](x0) = 0, [tt] = 0, [cons](x0, x1) = x0 + 1, [zeros] = 1, [0] = 0, [a__zeros] = 1 orientation: a__length#(cons(N,L)) = 0 >= 0 = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = 0 >= 0 = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(s(X)) = X >= X = mark#(X) mark#(cons(X1,X2)) = X1 + 1 >= X1 = mark#(X1) mark#(U52(X)) = X >= X = mark#(X) mark#(U42(X)) = X >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = 0 >= 0 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = 0 >= 0 = a__U52(a__isNatList(V2)) a__U52(tt()) = 0 >= 0 = tt() a__U61(tt(),L,N) = 1 >= 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 0 >= 0 = a__U51(a__isNat(V1),V2) a__length(nil()) = 1 >= 0 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 0 >= 0 = a__U11(mark(X)) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X)) = 0 >= 0 = a__U31(mark(X)) mark(U41(X1,X2)) = 0 >= 0 = a__U41(mark(X1),X2) mark(U42(X)) = X >= X = a__U42(mark(X)) mark(isNatIList(X)) = 0 >= 0 = a__isNatIList(X) mark(U51(X1,X2)) = 0 >= 0 = a__U51(mark(X1),X2) mark(U52(X)) = X >= X = a__U52(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(cons(X1,X2)) = X1 + 1 >= X1 + 1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X) = 0 >= 0 = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = 0 >= 0 = U31(X) a__U41(X1,X2) = 0 >= 0 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2) = 0 >= 0 = U51(X1,X2) a__U52(X) = X >= X = U52(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U61(X1,X2,X3) = 1 >= 1 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 1 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 1 >= 1 = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) mark#(s(X)) -> mark#(X) mark#(U52(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = x0, [mark#](x0) = x0, [a__U62#](x0, x1) = x1 + 1, [a__U61#](x0, x1, x2) = x1 + 1, [isNat](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [isNatList](x0) = x0, [U52](x0) = x0 + 1, [U51](x0, x1) = x1 + 1, [isNatIList](x0) = 0, [U42](x0) = x0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = 0, [nil] = 0, [length](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0 + 1, [a__U62](x0, x1) = 0, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = 0, [a__U52](x0) = x0 + 1, [a__isNatList](x0) = x0, [a__U51](x0, x1) = x1 + 1, [a__U42](x0) = x0, [a__isNatIList](x0) = 0, [a__U41](x0, x1) = 0, [a__U31](x0) = 0, [a__U21](x0) = x0, [a__U11](x0) = 0, [tt] = 0, [cons](x0, x1) = x1 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 1 orientation: a__length#(cons(N,L)) = L + 1 >= L + 1 = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = L + 1 >= L + 1 = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = L + 1 >= L + 1 = a__length#(mark(L)) mark#(s(X)) = X >= X = mark#(X) mark#(U52(X)) = X + 1 >= X = mark#(X) mark#(U42(X)) = X >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = 0 >= 0 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = V2 + 1 >= V2 + 1 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 0 = tt() a__U61(tt(),L,N) = 0 >= 0 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = V2 + 1 >= V2 + 1 = a__U51(a__isNat(V1),V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 1 >= 0 = a__U11(mark(X)) mark(U21(X)) = X + 1 >= X + 1 = a__U21(mark(X)) mark(U31(X)) = 1 >= 0 = a__U31(mark(X)) mark(U41(X1,X2)) = 1 >= 0 = a__U41(mark(X1),X2) mark(U42(X)) = X + 1 >= X + 1 = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 0 = a__isNatIList(X) mark(U51(X1,X2)) = X2 + 2 >= X2 + 1 = a__U51(mark(X1),X2) mark(U52(X)) = X + 2 >= X + 2 = a__U52(mark(X)) mark(isNatList(X)) = X + 1 >= X = a__isNatList(X) mark(U61(X1,X2,X3)) = 1 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 0 = a__U62(mark(X1),X2) mark(isNat(X)) = 1 >= 0 = a__isNat(X) mark(length(X)) = 1 >= 0 = a__length(mark(X)) mark(cons(X1,X2)) = X2 + 2 >= X2 + 1 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 1 >= 0 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 1 >= 0 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X) = 0 >= 0 = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = 0 >= 0 = U31(X) a__U41(X1,X2) = 0 >= 0 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2) = X2 + 1 >= X2 + 1 = U51(X1,X2) a__U52(X) = X + 1 >= X + 1 = U52(X) a__isNatList(X) = X >= X = isNatList(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 0 >= 0 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 0 >= 0 = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) mark#(s(X)) -> mark#(X) mark#(U42(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__length#](x0) = 0, [mark#](x0) = x0, [a__U62#](x0, x1) = 0, [a__U61#](x0, x1, x2) = 0, [isNat](x0) = 0, [U62](x0, x1) = x0, [U61](x0, x1, x2) = 0, [isNatList](x0) = x0, [U52](x0) = 0, [U51](x0, x1) = 1, [isNatIList](x0) = x0, [U42](x0) = x0 + 1, [U41](x0, x1) = x1, [U31](x0) = 0, [U21](x0) = x0, [U11](x0) = 0, [nil] = 0, [length](x0) = 0, [s](x0) = x0, [a__length](x0) = 0, [mark](x0) = x0 + 1, [a__U62](x0, x1) = x0, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = 0, [a__U52](x0) = 1, [a__isNatList](x0) = x0, [a__U51](x0, x1) = 1, [a__U42](x0) = x0 + 1, [a__isNatIList](x0) = x0, [a__U41](x0, x1) = x1 + 1, [a__U31](x0) = 0, [a__U21](x0) = x0, [a__U11](x0) = 0, [tt] = 0, [cons](x0, x1) = x1 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 1 orientation: a__length#(cons(N,L)) = 0 >= 0 = a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) = 0 >= 0 = a__U62#(a__isNat(N),L) a__U62#(tt(),L) = 0 >= 0 = a__length#(mark(L)) mark#(s(X)) = X >= X = mark#(X) mark#(U42(X)) = X + 1 >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = V2 + 1 >= V2 + 1 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 1 >= 0 = tt() a__U51(tt(),V2) = 1 >= 1 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 0 = tt() a__U61(tt(),L,N) = 0 >= 0 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 0 >= 0 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = V >= 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V2 + 1 >= V2 + 1 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = V2 + 1 >= 1 = a__U51(a__isNat(V1),V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = 0 >= 0 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 1 >= 0 = a__U11(mark(X)) mark(U21(X)) = X + 1 >= X + 1 = a__U21(mark(X)) mark(U31(X)) = 1 >= 0 = a__U31(mark(X)) mark(U41(X1,X2)) = X2 + 1 >= X2 + 1 = a__U41(mark(X1),X2) mark(U42(X)) = X + 2 >= X + 2 = a__U42(mark(X)) mark(isNatIList(X)) = X + 1 >= X = a__isNatIList(X) mark(U51(X1,X2)) = 2 >= 1 = a__U51(mark(X1),X2) mark(U52(X)) = 1 >= 1 = a__U52(mark(X)) mark(isNatList(X)) = X + 1 >= X = a__isNatList(X) mark(U61(X1,X2,X3)) = 1 >= 0 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X1 + 1 >= X1 + 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 1 >= 0 = a__isNat(X) mark(length(X)) = 1 >= 0 = a__length(mark(X)) mark(cons(X1,X2)) = X2 + 2 >= X2 + 1 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 1 >= 0 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 1 >= 0 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X) = 0 >= 0 = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = 0 >= 0 = U31(X) a__U41(X1,X2) = X2 + 1 >= X2 = U41(X1,X2) a__U42(X) = X + 1 >= X + 1 = U42(X) a__isNatIList(X) = X >= X = isNatIList(X) a__U51(X1,X2) = 1 >= 1 = U51(X1,X2) a__U52(X) = 1 >= 0 = U52(X) a__isNatList(X) = X >= X = isNatList(X) a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 0 >= 0 = length(X) problem: DPs: a__length#(cons(N,L)) -> a__U61#(a__isNatList(L),L,N) a__U61#(tt(),L,N) -> a__U62#(a__isNat(N),L) a__U62#(tt(),L) -> a__length#(mark(L)) mark#(s(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Open DPs: a__isNatIList#(cons(V1,V2)) -> a__U41#(a__isNat(V1),V2) a__U41#(tt(),V2) -> a__isNatIList#(V2) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatIList#](x0) = x0, [a__U41#](x0, x1) = x1, [isNat](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [isNatList](x0) = 0, [U52](x0) = 0, [U51](x0, x1) = x0, [isNatIList](x0) = 0, [U42](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 1, [U21](x0) = 0, [U11](x0) = 0, [nil] = 1, [length](x0) = x0 + 1, [s](x0) = 0, [a__length](x0) = x0 + 1, [mark](x0) = x0 + 1, [a__U62](x0, x1) = 1, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = 1, [a__U52](x0) = 0, [a__isNatList](x0) = 1, [a__U51](x0, x1) = x0, [a__U42](x0) = 1, [a__isNatIList](x0) = 1, [a__U41](x0, x1) = 1, [a__U31](x0) = 1, [a__U21](x0) = 0, [a__U11](x0) = 0, [tt] = 0, [cons](x0, x1) = x0 + x1 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 1 orientation: a__isNatIList#(cons(V1,V2)) = V1 + V2 + 1 >= V2 = a__U41#(a__isNat(V1),V2) a__U41#(tt(),V2) = V2 >= V2 = a__isNatIList#(V2) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 1 >= 0 = tt() a__U41(tt(),V2) = 1 >= 1 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 1 >= 0 = tt() a__U51(tt(),V2) = 0 >= 0 = a__U52(a__isNatList(V2)) a__U52(tt()) = 0 >= 0 = tt() a__U61(tt(),L,N) = 1 >= 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 1 >= 0 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 1 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1 >= 0 = a__U51(a__isNat(V1),V2) a__length(nil()) = 2 >= 0 = 0() a__length(cons(N,L)) = L + N + 2 >= 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 1 >= 0 = a__U11(mark(X)) mark(U21(X)) = 1 >= 0 = a__U21(mark(X)) mark(U31(X)) = 2 >= 1 = a__U31(mark(X)) mark(U41(X1,X2)) = 1 >= 1 = a__U41(mark(X1),X2) mark(U42(X)) = 1 >= 1 = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2)) = X1 + 1 >= X1 + 1 = a__U51(mark(X1),X2) mark(U52(X)) = 1 >= 0 = a__U52(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 1 >= 0 = a__isNat(X) mark(length(X)) = X + 2 >= X + 2 = a__length(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 1 >= 0 = tt() mark(s(X)) = 1 >= 0 = s(mark(X)) mark(nil()) = 2 >= 1 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X) = 0 >= 0 = U11(X) a__U21(X) = 0 >= 0 = U21(X) a__U31(X) = 1 >= 1 = U31(X) a__U41(X1,X2) = 1 >= 0 = U41(X1,X2) a__U42(X) = 1 >= 0 = U42(X) a__isNatIList(X) = 1 >= 0 = isNatIList(X) a__U51(X1,X2) = X1 >= X1 = U51(X1,X2) a__U52(X) = 0 >= 0 = U52(X) a__isNatList(X) = 1 >= 0 = isNatList(X) a__U61(X1,X2,X3) = 1 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 0 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = X + 1 >= X + 1 = length(X) problem: DPs: a__U41#(tt(),V2) -> a__isNatIList#(V2) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNatIList#](x0) = 0, [a__U41#](x0, x1) = x0 + 1, [isNat](x0) = 1, [U62](x0, x1) = 1, [U61](x0, x1, x2) = x0, [isNatList](x0) = 1, [U52](x0) = 1, [U51](x0, x1) = 1, [isNatIList](x0) = 1, [U42](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = 1, [U21](x0) = x0, [U11](x0) = 1, [nil] = 1, [length](x0) = 1, [s](x0) = 1, [a__length](x0) = 1, [mark](x0) = 1, [a__U62](x0, x1) = 1, [a__isNat](x0) = 1, [a__U61](x0, x1, x2) = x0, [a__U52](x0) = 1, [a__isNatList](x0) = 1, [a__U51](x0, x1) = 1, [a__U42](x0) = 1, [a__isNatIList](x0) = 1, [a__U41](x0, x1) = 1, [a__U31](x0) = 1, [a__U21](x0) = x0, [a__U11](x0) = 1, [tt] = 1, [cons](x0, x1) = 1, [zeros] = 1, [0] = 1, [a__zeros] = 1 orientation: a__U41#(tt(),V2) = 2 >= 0 = a__isNatIList#(V2) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 1 >= 1 = tt() a__U21(tt()) = 1 >= 1 = tt() a__U31(tt()) = 1 >= 1 = tt() a__U41(tt(),V2) = 1 >= 1 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 1 >= 1 = tt() a__U51(tt(),V2) = 1 >= 1 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 1 = tt() a__U61(tt(),L,N) = 1 >= 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 1 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__isNat(V1),V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 1 >= 1 = a__U11(mark(X)) mark(U21(X)) = 1 >= 1 = a__U21(mark(X)) mark(U31(X)) = 1 >= 1 = a__U31(mark(X)) mark(U41(X1,X2)) = 1 >= 1 = a__U41(mark(X1),X2) mark(U42(X)) = 1 >= 1 = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2) mark(U52(X)) = 1 >= 1 = a__U52(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(cons(X1,X2)) = 1 >= 1 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 1 >= 1 = tt() mark(s(X)) = 1 >= 1 = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X) = 1 >= 1 = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = 1 >= 1 = U31(X) a__U41(X1,X2) = 1 >= 1 = U41(X1,X2) a__U42(X) = 1 >= 1 = U42(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2) = 1 >= 1 = U51(X1,X2) a__U52(X) = 1 >= 1 = U52(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 1 = U62(X1,X2) a__isNat(X) = 1 >= 1 = isNat(X) a__length(X) = 1 >= 1 = length(X) problem: DPs: TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Qed DPs: a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) a__U51#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNat#(length(V1)) -> a__isNatList#(V1) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNat#](x0) = x0 + 1, [a__isNatList#](x0) = x0, [a__U51#](x0, x1) = x1, [isNat](x0) = 0, [U62](x0, x1) = x1, [U61](x0, x1, x2) = x1 + x2, [isNatList](x0) = x0, [U52](x0) = 0, [U51](x0, x1) = x0 + x1 + 1, [isNatIList](x0) = 0, [U42](x0) = x0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = 0, [U11](x0) = 0, [nil] = 0, [length](x0) = x0, [s](x0) = x0, [a__length](x0) = x0, [mark](x0) = x0 + 1, [a__U62](x0, x1) = x1 + 1, [a__isNat](x0) = 0, [a__U61](x0, x1, x2) = x1 + x2 + 1, [a__U52](x0) = 1, [a__isNatList](x0) = x0, [a__U51](x0, x1) = x0 + x1 + 1, [a__U42](x0) = x0, [a__isNatIList](x0) = 0, [a__U41](x0, x1) = 0, [a__U31](x0) = 0, [a__U21](x0) = 0, [a__U11](x0) = 0, [tt] = 0, [cons](x0, x1) = x0 + x1 + 1, [zeros] = 0, [0] = 0, [a__zeros] = 1 orientation: a__isNatList#(cons(V1,V2)) = V1 + V2 + 1 >= V2 = a__U51#(a__isNat(V1),V2) a__U51#(tt(),V2) = V2 >= V2 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = V1 + V2 + 1 >= V1 + 1 = a__isNat#(V1) a__isNat#(s(V1)) = V1 + 1 >= V1 + 1 = a__isNat#(V1) a__isNat#(length(V1)) = V1 + 1 >= V1 = a__isNatList#(V1) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 0 >= 0 = tt() a__U21(tt()) = 0 >= 0 = tt() a__U31(tt()) = 0 >= 0 = tt() a__U41(tt(),V2) = 0 >= 0 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = V2 + 1 >= 1 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 0 = tt() a__U61(tt(),L,N) = L + N + 1 >= L + 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = L + 1 >= L + 1 = s(a__length(mark(L))) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = 0 >= 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = 0 >= 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 0 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 0 >= 0 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 + 1 >= V2 + 1 = a__U51(a__isNat(V1),V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = L + N + 1 >= L + N + 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 1 >= 0 = a__U11(mark(X)) mark(U21(X)) = 1 >= 0 = a__U21(mark(X)) mark(U31(X)) = 1 >= 0 = a__U31(mark(X)) mark(U41(X1,X2)) = 1 >= 0 = a__U41(mark(X1),X2) mark(U42(X)) = X + 1 >= X + 1 = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 0 = a__isNatIList(X) mark(U51(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = a__U51(mark(X1),X2) mark(U52(X)) = 1 >= 1 = a__U52(mark(X)) mark(isNatList(X)) = X + 1 >= X = a__isNatList(X) mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = X2 + 1 >= X2 + 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 1 >= 0 = a__isNat(X) mark(length(X)) = X + 1 >= X + 1 = a__length(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() mark(tt()) = 1 >= 0 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 1 >= 0 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X) = 0 >= 0 = U11(X) a__U21(X) = 0 >= 0 = U21(X) a__U31(X) = 0 >= 0 = U31(X) a__U41(X1,X2) = 0 >= 0 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2) a__U52(X) = 1 >= 0 = U52(X) a__isNatList(X) = X >= X = isNatList(X) a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 = U61(X1,X2,X3) a__U62(X1,X2) = X2 + 1 >= X2 = U62(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = X >= X = length(X) problem: DPs: a__U51#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__isNatList#](x0) = 0, [a__U51#](x0, x1) = x0 + 1, [isNat](x0) = 0, [U62](x0, x1) = 0, [U61](x0, x1, x2) = 0, [isNatList](x0) = x0, [U52](x0) = 0, [U51](x0, x1) = 0, [isNatIList](x0) = 0, [U42](x0) = 0, [U41](x0, x1) = 0, [U31](x0) = 0, [U21](x0) = 0, [U11](x0) = 0, [nil] = 1, [length](x0) = 0, [s](x0) = x0, [a__length](x0) = 1, [mark](x0) = x0 + 1, [a__U62](x0, x1) = 1, [a__isNat](x0) = 1, [a__U61](x0, x1, x2) = 1, [a__U52](x0) = 1, [a__isNatList](x0) = x0, [a__U51](x0, x1) = 1, [a__U42](x0) = 1, [a__isNatIList](x0) = 1, [a__U41](x0, x1) = 1, [a__U31](x0) = 1, [a__U21](x0) = 1, [a__U11](x0) = 1, [tt] = 1, [cons](x0, x1) = 1, [zeros] = 0, [0] = 1, [a__zeros] = 1 orientation: a__U51#(tt(),V2) = 2 >= 0 = a__isNatList#(V2) a__isNatList#(cons(V1,V2)) = 0 >= 0 = a__isNat#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 1 >= 1 = tt() a__U21(tt()) = 1 >= 1 = tt() a__U31(tt()) = 1 >= 1 = tt() a__U41(tt(),V2) = 1 >= 1 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 1 >= 1 = tt() a__U51(tt(),V2) = 1 >= 1 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 1 = tt() a__U61(tt(),L,N) = 1 >= 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 1 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__isNat(V1),V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 1 >= 1 = a__U11(mark(X)) mark(U21(X)) = 1 >= 1 = a__U21(mark(X)) mark(U31(X)) = 1 >= 1 = a__U31(mark(X)) mark(U41(X1,X2)) = 1 >= 1 = a__U41(mark(X1),X2) mark(U42(X)) = 1 >= 1 = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2) mark(U52(X)) = 1 >= 1 = a__U52(mark(X)) mark(isNatList(X)) = X + 1 >= X = a__isNatList(X) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(cons(X1,X2)) = 2 >= 1 = cons(mark(X1),X2) mark(0()) = 2 >= 1 = 0() mark(tt()) = 2 >= 1 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(nil()) = 2 >= 1 = nil() a__zeros() = 1 >= 0 = zeros() a__U11(X) = 1 >= 0 = U11(X) a__U21(X) = 1 >= 0 = U21(X) a__U31(X) = 1 >= 0 = U31(X) a__U41(X1,X2) = 1 >= 0 = U41(X1,X2) a__U42(X) = 1 >= 0 = U42(X) a__isNatIList(X) = 1 >= 0 = isNatIList(X) a__U51(X1,X2) = 1 >= 0 = U51(X1,X2) a__U52(X) = 1 >= 0 = U52(X) a__isNatList(X) = X >= X = isNatList(X) a__U61(X1,X2,X3) = 1 >= 0 = U61(X1,X2,X3) a__U62(X1,X2) = 1 >= 0 = U62(X1,X2) a__isNat(X) = 1 >= 0 = isNat(X) a__length(X) = 1 >= 0 = length(X) problem: DPs: a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Matrix Interpretation Processor: dimension: 1 interpretation: [a__isNat#](x0) = 0, [a__isNatList#](x0) = x0 + 1, [isNat](x0) = 1, [U62](x0, x1) = x0, [U61](x0, x1, x2) = 1, [isNatList](x0) = 1, [U52](x0) = 1, [U51](x0, x1) = 0, [isNatIList](x0) = 1, [U42](x0) = 1, [U41](x0, x1) = 1, [U31](x0) = 1, [U21](x0) = 1, [U11](x0) = 1, [nil] = 1, [length](x0) = 1, [s](x0) = 1, [a__length](x0) = 1, [mark](x0) = 1, [a__U62](x0, x1) = x0, [a__isNat](x0) = 1, [a__U61](x0, x1, x2) = 1, [a__U52](x0) = 1, [a__isNatList](x0) = 1, [a__U51](x0, x1) = x0, [a__U42](x0) = 1, [a__isNatIList](x0) = 1, [a__U41](x0, x1) = 1, [a__U31](x0) = 1, [a__U21](x0) = 1, [a__U11](x0) = 1, [tt] = 1, [cons](x0, x1) = 1, [zeros] = 1, [0] = 1, [a__zeros] = 1 orientation: a__isNatList#(cons(V1,V2)) = 2 >= 0 = a__isNat#(V1) a__isNat#(s(V1)) = 0 >= 0 = a__isNat#(V1) a__zeros() = 1 >= 1 = cons(0(),zeros()) a__U11(tt()) = 1 >= 1 = tt() a__U21(tt()) = 1 >= 1 = tt() a__U31(tt()) = 1 >= 1 = tt() a__U41(tt(),V2) = 1 >= 1 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 1 >= 1 = tt() a__U51(tt(),V2) = 1 >= 1 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 1 = tt() a__U61(tt(),L,N) = 1 >= 1 = a__U62(a__isNat(N),L) a__U62(tt(),L) = 1 >= 1 = s(a__length(mark(L))) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 1 >= 1 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 1 >= 1 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 1 = tt() a__isNatIList(cons(V1,V2)) = 1 >= 1 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = 1 >= 1 = a__U51(a__isNat(V1),V2) a__length(nil()) = 1 >= 1 = 0() a__length(cons(N,L)) = 1 >= 1 = a__U61(a__isNatList(L),L,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = 1 >= 1 = a__U11(mark(X)) mark(U21(X)) = 1 >= 1 = a__U21(mark(X)) mark(U31(X)) = 1 >= 1 = a__U31(mark(X)) mark(U41(X1,X2)) = 1 >= 1 = a__U41(mark(X1),X2) mark(U42(X)) = 1 >= 1 = a__U42(mark(X)) mark(isNatIList(X)) = 1 >= 1 = a__isNatIList(X) mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2) mark(U52(X)) = 1 >= 1 = a__U52(mark(X)) mark(isNatList(X)) = 1 >= 1 = a__isNatList(X) mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) = 1 >= 1 = a__U62(mark(X1),X2) mark(isNat(X)) = 1 >= 1 = a__isNat(X) mark(length(X)) = 1 >= 1 = a__length(mark(X)) mark(cons(X1,X2)) = 1 >= 1 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 1 >= 1 = tt() mark(s(X)) = 1 >= 1 = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X) = 1 >= 1 = U11(X) a__U21(X) = 1 >= 1 = U21(X) a__U31(X) = 1 >= 1 = U31(X) a__U41(X1,X2) = 1 >= 1 = U41(X1,X2) a__U42(X) = 1 >= 1 = U42(X) a__isNatIList(X) = 1 >= 1 = isNatIList(X) a__U51(X1,X2) = X1 >= 0 = U51(X1,X2) a__U52(X) = 1 >= 1 = U52(X) a__isNatList(X) = 1 >= 1 = isNatList(X) a__U61(X1,X2,X3) = 1 >= 1 = U61(X1,X2,X3) a__U62(X1,X2) = X1 >= X1 = U62(X1,X2) a__isNat(X) = 1 >= 1 = isNat(X) a__length(X) = 1 >= 1 = length(X) problem: DPs: a__isNat#(s(V1)) -> a__isNat#(V1) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),L,N) -> a__U62(a__isNat(N),L) a__U62(tt(),L) -> s(a__length(mark(L))) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U61(a__isNatList(L),L,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3) mark(U62(X1,X2)) -> a__U62(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2,X3) -> U61(X1,X2,X3) a__U62(X1,X2) -> U62(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) Open