YES 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) 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(),V2) -> a__isNatIList#(V2) a__U61#(tt(),V2) -> a__U62#(a__isNatIList(V2)) a__U71#(tt(),L,N) -> a__isNat#(N) a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) a__U72#(tt(),L) -> mark#(L) a__U72#(tt(),L) -> a__length#(mark(L)) a__U91#(tt(),IL,M,N) -> a__isNat#(M) a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) -> a__isNat#(N) a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) -> mark#(N) 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__isNatList#(take(V1,V2)) -> a__isNat#(V1) a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__U81#(a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,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)) -> mark#(X1) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U62(X)) -> mark#(X) mark#(U62(X)) -> a__U62#(mark(X)) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U71(X1,X2,X3)) -> a__U71#(mark(X1),X2,X3) mark#(U72(X1,X2)) -> mark#(X1) mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> a__U81#(mark(X)) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) -> mark#(X1) mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) -> mark#(X1) mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) TDG 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(),V2) -> a__isNatIList#(V2) a__U61#(tt(),V2) -> a__U62#(a__isNatIList(V2)) a__U71#(tt(),L,N) -> a__isNat#(N) a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) a__U72#(tt(),L) -> mark#(L) a__U72#(tt(),L) -> a__length#(mark(L)) a__U91#(tt(),IL,M,N) -> a__isNat#(M) a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) -> a__isNat#(N) a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) -> mark#(N) 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__isNatList#(take(V1,V2)) -> a__isNat#(V1) a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2) a__length#(cons(N,L)) -> a__isNatList#(L) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) a__take#(0(),IL) -> a__isNatIList#(IL) a__take#(0(),IL) -> a__U81#(a__isNatIList(IL)) a__take#(s(M),cons(N,IL)) -> a__isNatIList#(IL) a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,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)) -> mark#(X1) mark#(U61(X1,X2)) -> a__U61#(mark(X1),X2) mark#(U62(X)) -> mark#(X) mark#(U62(X)) -> a__U62#(mark(X)) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U71(X1,X2,X3)) -> a__U71#(mark(X1),X2,X3) mark#(U72(X1,X2)) -> mark#(X1) mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2) mark#(isNat(X)) -> a__isNat#(X) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U81(X)) -> a__U81#(mark(X)) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) -> mark#(X1) mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) -> mark#(X1) mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) graph: ... SCC Processor: #sccs: 2 #rules: 47 #arcs: 1012/5476 DPs: a__take#(s(M),cons(N,IL)) -> a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) -> mark#(N) mark#(U11(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) mark#(U62(X)) -> mark#(X) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U71(X1,X2,X3)) -> a__U71#(mark(X1),X2,X3) a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) a__U72#(tt(),L) -> mark#(L) mark#(U72(X1,X2)) -> mark#(X1) mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2) a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) mark#(length(X)) -> mark#(X) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) -> mark#(X1) mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) -> mark#(X1) mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dim=3 usable rules: 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) interpretation: [a__take#](x0, x1) = [0 1 0]x1 + [1], [a__U93#](x0, x1, x2, x3) = [1 0 0]x0 + [0 1 0]x3, [a__U92#](x0, x1, x2, x3) = [1 0 0]x0 + [0 1 0]x3, [a__U91#](x0, x1, x2, x3) = [1 0 0]x0 + [0 1 1]x1 + [0 1 0]x3, [a__length#](x0) = [0 1 0]x0 + [1], [mark#](x0) = [0 1 0]x0, [a__U72#](x0, x1) = [0 1 0]x0 + [0 1 0]x1, [a__U71#](x0, x1, x2) = [0 1 0]x0 + [0 1 0]x1, [0 0 0] [0 0 0] [0 0 0] [0 0 0] [U93](x0, x1, x2, x3) = [1 1 1]x0 + [0 1 1]x1 + [0 1 1]x2 + [0 1 0]x3 [0 0 1] [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [0 0 0] [0 0 0] [U92](x0, x1, x2, x3) = [0 1 1]x0 + [0 1 1]x1 + [0 1 1]x2 + [0 1 0]x3 [0 0 0] [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [0 0 0] [0 0 0] [U91](x0, x1, x2, x3) = [0 1 1]x0 + [0 1 1]x1 + [0 1 1]x2 + [0 1 0]x3 [0 0 0] [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [U81](x0) = [0 1 0]x0 [0 0 0] , [0] [isNat](x0) = [1] [0], [0 0 0] [0 0 0] [U72](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [U71](x0, x1, x2) = [0 1 0]x0 + [0 1 1]x1 [0 0 0] [0 0 0] , [0 0 0] [U62](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [U61](x0, x1) = [0 1 0]x0 [0 0 0] , [0] [isNatList](x0) = [1] [0], [0 0 0] [U52](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [U51](x0, x1) = [1 1 1]x0 [0 0 0] , [0] [isNatIList](x0) = [1] [0], [0 0 0] [U42](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [U41](x0, x1) = [0 1 0]x0 [0 0 0] , [0 0 0] [U31](x0) = [1 1 1]x0 [0 0 0] , [0 0 0] [U21](x0) = [1 1 1]x0 [0 0 0] , [0 0 0] [U11](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [0 0 0] [0] [a__take](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [0 0 0] [0] [length](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [0 0 0] [0 0 0] [0] [take](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 0 0] [0 0 0] [0 0 0] [0 0 0] [a__U93](x0, x1, x2, x3) = [1 1 1]x0 + [0 1 1]x1 + [0 1 1]x2 + [0 1 0]x3 [0 0 1] [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [0 0 0] [0 0 0] [a__U92](x0, x1, x2, x3) = [0 1 1]x0 + [0 1 1]x1 + [0 1 1]x2 + [0 1 0]x3 [0 0 1] [0 0 0] [0 0 0] [0 0 0] , [0 0 0] [0 0 0] [0 0 0] [0 0 0] [a__U91](x0, x1, x2, x3) = [1 1 1]x0 + [0 1 1]x1 + [0 1 1]x2 + [0 1 0]x3 [0 0 1] [0 0 0] [0 0 0] [0 0 0] , [0] [nil] = [0] [0], [0 0 0] [a__U81](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [s](x0) = [0 1 1]x0 [0 0 0] , [0 0 0] [0] [a__length](x0) = [0 1 0]x0 + [1] [0 0 1] [0], [0 0 1] [mark](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [0 0 0] [a__U72](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [a__isNat](x0) = [1] [0], [0 0 0] [0 0 0] [a__U71](x0, x1, x2) = [0 1 0]x0 + [0 1 1]x1 [0 0 1] [0 0 0] , [0 0 0] [a__U62](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [a__U61](x0, x1) = [0 1 0]x0 [0 0 0] , [0 0 0] [a__U52](x0) = [0 1 0]x0 [0 0 0] , [0] [a__isNatList](x0) = [1] [0], [0 0 0] [a__U51](x0, x1) = [1 1 1]x0 [0 0 0] , [0 0 0] [a__U42](x0) = [0 1 0]x0 [0 0 0] , [0] [a__isNatIList](x0) = [1] [0], [0 0 0] [a__U41](x0, x1) = [0 1 0]x0 [0 0 0] , [0 0 0] [a__U31](x0) = [1 1 1]x0 [0 0 0] , [0 0 0] [a__U21](x0) = [1 1 1]x0 [0 0 0] , [0 0 0] [a__U11](x0) = [0 1 0]x0 [0 0 0] , [0] [tt] = [1] [0], [0 0 0] [0 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 1 1]x1 [0 0 0] [0 0 0] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0] orientation: a__take#(s(M),cons(N,IL)) = [0 1 1]IL + [0 1 0]N + [1] >= [0 1 1]IL + [0 1 0]N = a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) = [0 1 1]IL + [0 1 0]N >= [0 1 0]N = a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) = [0 1 0]N >= [0 1 0]N = a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) = [0 1 0]N >= [0 1 0]N = mark#(N) mark#(U11(X)) = [0 1 0]X >= [0 1 0]X = mark#(X) mark#(U21(X)) = [1 1 1]X >= [0 1 0]X = mark#(X) mark#(U31(X)) = [1 1 1]X >= [0 1 0]X = mark#(X) mark#(U41(X1,X2)) = [0 1 0]X1 >= [0 1 0]X1 = mark#(X1) mark#(U42(X)) = [0 1 0]X >= [0 1 0]X = mark#(X) mark#(U51(X1,X2)) = [1 1 1]X1 >= [0 1 0]X1 = mark#(X1) mark#(U52(X)) = [0 1 0]X >= [0 1 0]X = mark#(X) mark#(U61(X1,X2)) = [0 1 0]X1 >= [0 1 0]X1 = mark#(X1) mark#(U62(X)) = [0 1 0]X >= [0 1 0]X = mark#(X) mark#(U71(X1,X2,X3)) = [0 1 0]X1 + [0 1 1]X2 >= [0 1 0]X1 = mark#(X1) mark#(U71(X1,X2,X3)) = [0 1 0]X1 + [0 1 1]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__U71#(mark(X1),X2,X3) a__U71#(tt(),L,N) = [0 1 0]L + [1] >= [0 1 0]L + [1] = a__U72#(a__isNat(N),L) a__U72#(tt(),L) = [0 1 0]L + [1] >= [0 1 0]L = mark#(L) mark#(U72(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 = mark#(X1) mark#(U72(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__U72#(mark(X1),X2) a__U72#(tt(),L) = [0 1 0]L + [1] >= [0 1 0]L + [1] = a__length#(mark(L)) a__length#(cons(N,L)) = [0 1 1]L + [0 1 0]N + [1] >= [0 1 0]L + [1] = a__U71#(a__isNatList(L),L,N) mark#(length(X)) = [0 1 0]X + [1] >= [0 1 0]X = mark#(X) mark#(length(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__length#(mark(X)) mark#(U81(X)) = [0 1 0]X >= [0 1 0]X = mark#(X) mark#(U91(X1,X2,X3,X4)) = [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 1 0]X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 0 1]X1 + [0 1 1]X2 + [0 1 0]X4 = a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) = [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 1 0]X1 = mark#(X1) mark#(U92(X1,X2,X3,X4)) = [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 0 1]X1 + [0 1 0]X4 = a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) = [1 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 1 0]X1 = mark#(X1) mark#(U93(X1,X2,X3,X4)) = [1 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 0 1]X1 + [0 1 0]X4 = a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X2 = mark#(X2) mark#(take(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 = mark#(X1) mark#(take(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X2 + [1] = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = [0 1 0]X1 + [0 1 1]X2 >= [0 1 0]X1 = mark#(X1) mark#(s(X)) = [0 1 1]X >= [0 1 0]X = mark#(X) [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [0] [0] a__U11(tt()) = [1] >= [1] = tt() [0] [0] [0] [0] a__U21(tt()) = [1] >= [1] = tt() [0] [0] [0] [0] a__U31(tt()) = [1] >= [1] = tt() [0] [0] [0] [0] a__U41(tt(),V2) = [1] >= [1] = a__U42(a__isNatIList(V2)) [0] [0] [0] [0] a__U42(tt()) = [1] >= [1] = tt() [0] [0] [0] [0] a__U51(tt(),V2) = [1] >= [1] = a__U52(a__isNatList(V2)) [0] [0] [0] [0] a__U52(tt()) = [1] >= [1] = tt() [0] [0] [0] [0] a__U61(tt(),V2) = [1] >= [1] = a__U62(a__isNatIList(V2)) [0] [0] [0] [0] a__U62(tt()) = [1] >= [1] = tt() [0] [0] [0 0 0] [0] [0 0 0] [0] a__U71(tt(),L,N) = [0 1 1]L + [1] >= [0 1 0]L + [1] = a__U72(a__isNat(N),L) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] a__U72(tt(),L) = [0 1 0]L + [1] >= [0 1 0]L + [1] = s(a__length(mark(L))) [0 0 0] [0] [0 0 0] [0] [0] [0] a__U81(tt()) = [1] >= [0] = nil() [0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] a__U91(tt(),IL,M,N) = [0 1 1]IL + [0 1 1]M + [0 1 0]N + [1] >= [0 1 1]IL + [0 1 1]M + [0 1 0]N + [1] = a__U92(a__isNat(M),IL,M,N) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] a__U92(tt(),IL,M,N) = [0 1 1]IL + [0 1 1]M + [0 1 0]N + [1] >= [0 1 1]IL + [0 1 1]M + [0 1 0]N + [1] = a__U93(a__isNat(N),IL,M,N) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] a__U93(tt(),IL,M,N) = [0 1 1]IL + [0 1 1]M + [0 1 0]N + [1] >= [0 1 0]IL + [0 1 0]M + [0 1 0]N + [1] = cons(mark(N),take(M,IL)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0] [0] a__isNat(0()) = [1] >= [1] = tt() [0] [0] [0] [0] a__isNat(length(V1)) = [1] >= [1] = a__U11(a__isNatList(V1)) [0] [0] [0] [0] a__isNat(s(V1)) = [1] >= [1] = a__U21(a__isNat(V1)) [0] [0] [0] [0] a__isNatIList(V) = [1] >= [1] = a__U31(a__isNatList(V)) [0] [0] [0] [0] a__isNatIList(zeros()) = [1] >= [1] = tt() [0] [0] [0] [0] a__isNatIList(cons(V1,V2)) = [1] >= [1] = a__U41(a__isNat(V1),V2) [0] [0] [0] [0] a__isNatList(nil()) = [1] >= [1] = tt() [0] [0] [0] [0] a__isNatList(cons(V1,V2)) = [1] >= [1] = a__U51(a__isNat(V1),V2) [0] [0] [0] [0] a__isNatList(take(V1,V2)) = [1] >= [1] = a__U61(a__isNat(V1),V2) [0] [0] [0] [0] a__length(nil()) = [1] >= [0] = 0() [0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0] a__length(cons(N,L)) = [0 1 1]L + [0 1 0]N + [1] >= [0 1 1]L + [1] = a__U71(a__isNatList(L),L,N) [0 0 0] [0 0 0] [0] [0 0 0] [0] [0 0 0] [0] [0] a__take(0(),IL) = [0 1 0]IL + [1] >= [1] = a__U81(a__isNatIList(IL)) [0 0 0] [0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] a__take(s(M),cons(N,IL)) = [0 1 1]IL + [0 1 1]M + [0 1 0]N + [1] >= [0 1 1]IL + [0 1 1]M + [0 1 0]N + [1] = a__U91(a__isNatIList(IL),IL,M,N) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [0 0 0] [0 0 0] mark(U11(X)) = [0 1 0]X >= [0 1 0]X = a__U11(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U21(X)) = [1 1 1]X >= [0 1 1]X = a__U21(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U31(X)) = [1 1 1]X >= [0 1 1]X = a__U31(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U41(X1,X2)) = [0 1 0]X1 >= [0 1 0]X1 = a__U41(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U42(X)) = [0 1 0]X >= [0 1 0]X = a__U42(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(isNatIList(X)) = [1] >= [1] = a__isNatIList(X) [0] [0] [0 0 0] [0 0 0] mark(U51(X1,X2)) = [1 1 1]X1 >= [0 1 1]X1 = a__U51(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U52(X)) = [0 1 0]X >= [0 1 0]X = a__U52(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(isNatList(X)) = [1] >= [1] = a__isNatList(X) [0] [0] [0 0 0] [0 0 0] mark(U61(X1,X2)) = [0 1 0]X1 >= [0 1 0]X1 = a__U61(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U62(X)) = [0 1 0]X >= [0 1 0]X = a__U62(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U71(X1,X2,X3)) = [0 1 0]X1 + [0 1 1]X2 >= [0 1 0]X1 + [0 1 1]X2 = a__U71(mark(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U72(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = a__U72(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(isNat(X)) = [1] >= [1] = a__isNat(X) [0] [0] [0 0 1] [0] [0 0 0] [0] mark(length(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__length(mark(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0 0 0] mark(U81(X)) = [0 1 0]X >= [0 1 0]X = a__U81(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U91(X1,X2,X3,X4)) = [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 = a__U91(mark(X1),X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U92(X1,X2,X3,X4)) = [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 1 0]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 = a__U92(mark(X1),X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 0] mark(U93(X1,X2,X3,X4)) = [1 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 = a__U93(mark(X1),X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] mark(take(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = a__take(mark(X1),mark(X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(cons(X1,X2)) = [0 1 0]X1 + [0 1 1]X2 >= [0 1 0]X1 + [0 1 1]X2 = cons(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0] [0] mark(tt()) = [1] >= [1] = tt() [0] [0] [0 0 0] [0 0 0] mark(s(X)) = [0 1 1]X >= [0 1 0]X = s(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [0 0 0] [0 0 0] a__U11(X) = [0 1 0]X >= [0 1 0]X = U11(X) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U21(X) = [1 1 1]X >= [1 1 1]X = U21(X) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U31(X) = [1 1 1]X >= [1 1 1]X = U31(X) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U41(X1,X2) = [0 1 0]X1 >= [0 1 0]X1 = U41(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U42(X) = [0 1 0]X >= [0 1 0]X = U42(X) [0 0 0] [0 0 0] [0] [0] a__isNatIList(X) = [1] >= [1] = isNatIList(X) [0] [0] [0 0 0] [0 0 0] a__U51(X1,X2) = [1 1 1]X1 >= [1 1 1]X1 = U51(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U52(X) = [0 1 0]X >= [0 1 0]X = U52(X) [0 0 0] [0 0 0] [0] [0] a__isNatList(X) = [1] >= [1] = isNatList(X) [0] [0] [0 0 0] [0 0 0] a__U61(X1,X2) = [0 1 0]X1 >= [0 1 0]X1 = U61(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U62(X) = [0 1 0]X >= [0 1 0]X = U62(X) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U71(X1,X2,X3) = [0 1 0]X1 + [0 1 1]X2 >= [0 1 0]X1 + [0 1 1]X2 = U71(X1,X2,X3) [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U72(X1,X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = U72(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] a__isNat(X) = [1] >= [1] = isNat(X) [0] [0] [0 0 0] [0] [0 0 0] [0] a__length(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = length(X) [0 0 1] [0] [0 0 1] [0] [0 0 0] [0 0 0] a__U81(X) = [0 1 0]X >= [0 1 0]X = U81(X) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U91(X1,X2,X3,X4) = [1 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 = U91(X1,X2,X3,X4) [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U92(X1,X2,X3,X4) = [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [0 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 = U92(X1,X2,X3,X4) [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U93(X1,X2,X3,X4) = [1 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 >= [1 1 1]X1 + [0 1 1]X2 + [0 1 1]X3 + [0 1 0]X4 = U93(X1,X2,X3,X4) [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] a__take(X1,X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = take(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] problem: DPs: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) -> mark#(N) mark#(U11(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) mark#(U62(X)) -> mark#(X) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U71(X1,X2,X3)) -> a__U71#(mark(X1),X2,X3) a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) mark#(U72(X1,X2)) -> mark#(X1) mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2) a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) -> mark#(X1) mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) -> mark#(X1) mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Restore Modifier: DPs: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) -> mark#(N) mark#(U11(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) mark#(U62(X)) -> mark#(X) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U71(X1,X2,X3)) -> a__U71#(mark(X1),X2,X3) a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) mark#(U72(X1,X2)) -> mark#(X1) mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2) a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) -> mark#(X1) mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) -> mark#(X1) mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) SCC Processor: #sccs: 2 #rules: 26 #arcs: 609/900 DPs: a__U93#(tt(),IL,M,N) -> mark#(N) mark#(U11(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) mark#(U62(X)) -> mark#(X) mark#(U71(X1,X2,X3)) -> mark#(X1) mark#(U72(X1,X2)) -> mark#(X1) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N) mark#(U92(X1,X2,X3,X4)) -> mark#(X1) mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) -> mark#(X1) mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(mark#) = 0 pi(a__U91#) = 3 pi(a__U92#) = 3 pi(a__U93#) = 3 problem: DPs: a__U93#(tt(),IL,M,N) -> mark#(N) a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 365/9 DPs: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dim=3 usable rules: 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) interpretation: [a__length#](x0) = [0 0 1]x0 + [1], [a__U72#](x0, x1) = [1 0 0]x1 + [1], [a__U71#](x0, x1, x2) = [1 0 0]x0 + [1 0 0]x1 + [1], [1 0 0] [1 0 0] [1] [U93](x0, x1, x2, x3) = [0 0 0]x0 + [0 0 0]x2 + [0] [1 0 0] [0 0 0] [0], [1 1 0] [1 0 0] [1] [U92](x0, x1, x2, x3) = [0 0 0]x0 + [0 0 0]x2 + [0] [1 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [1] [U91](x0, x1, x2, x3) = [0 0 0]x0 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [U81](x0) = [0 0 0]x0 [1 0 0] , [1] [isNat](x0) = [0] [0], [1 0 0] [1] [U72](x0, x1) = [0 0 0]x1 + [0] [0 0 0] [0], [1 1 0] [1 0 0] [U71](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [U62](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U61](x0, x1) = [0 0 0]x0 [0 0 0] , [0 0 1] [isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U52](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [U51](x0, x1) = [0 0 0]x1 [0 0 0] , [1] [isNatIList](x0) = [0] [0], [1] [U42](x0) = [0] [0], [1 0 0] [U41](x0, x1) = [0 0 0]x0 [0 0 0] , [1] [U31](x0) = [0] [0], [1] [U21](x0) = [0] [0], [1] [U11](x0) = [0] [0], [1 0 0] [1] [a__take](x0, x1) = [0 0 0]x0 + [0] [1 0 0] [1], [1 0 0] [length](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1] [take](x0, x1) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [1] [a__U93](x0, x1, x2, x3) = [0 0 0]x0 + [0 0 0]x2 + [0] [1 0 0] [1 0 0] [1], [1 1 0] [1 0 0] [1] [a__U92](x0, x1, x2, x3) = [0 0 0]x0 + [0 0 0]x2 + [0] [1 0 0] [1 0 0] [1], [1 0 0] [1 0 0] [1] [a__U91](x0, x1, x2, x3) = [0 0 0]x0 + [0 0 0]x2 + [0] [1 0 0] [1 0 0] [1], [1] [nil] = [0] [1], [1 0 0] [a__U81](x0) = [0 0 0]x0 [1 0 0] , [1 1 0] [1] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [a__length](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [mark](x0) = [0 0 0]x0 [1 0 0] , [1 0 0] [1] [a__U72](x0, x1) = [0 0 0]x1 + [0] [0 0 0] [0], [1] [a__isNat](x0) = [0] [0], [1 1 0] [1 0 0] [a__U71](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 0] [a__U62](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [a__U61](x0, x1) = [0 0 0]x0 [0 1 0] , [1 0 0] [a__U52](x0) = [0 0 0]x0 [0 0 0] , [0 0 1] [a__isNatList](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [a__U51](x0, x1) = [0 0 0]x1 [0 0 0] , [1] [a__U42](x0) = [0] [0], [1] [a__isNatIList](x0) = [0] [0], [1 0 0] [a__U41](x0, x1) = [0 0 0]x0 [0 0 0] , [1] [a__U31](x0) = [0] [0], [1] [a__U21](x0) = [0] [0], [1] [a__U11](x0) = [0] [0], [1] [tt] = [0] [0], [1 0 1] [cons](x0, x1) = [0 0 0]x1 [1 0 1] , [0] [zeros] = [0] [0], [0] [0] = [0] [0], [0] [a__zeros] = [0] [0] orientation: a__U71#(tt(),L,N) = [1 0 0]L + [2] >= [1 0 0]L + [1] = a__U72#(a__isNat(N),L) a__U72#(tt(),L) = [1 0 0]L + [1] >= [1 0 0]L + [1] = a__length#(mark(L)) a__length#(cons(N,L)) = [1 0 1]L + [1] >= [1 0 1]L + [1] = a__U71#(a__isNatList(L),L,N) [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [1] [1] a__U11(tt()) = [0] >= [0] = tt() [0] [0] [1] [1] a__U21(tt()) = [0] >= [0] = tt() [0] [0] [1] [1] a__U31(tt()) = [0] >= [0] = tt() [0] [0] [1] [1] a__U41(tt(),V2) = [0] >= [0] = a__U42(a__isNatIList(V2)) [0] [0] [1] [1] a__U42(tt()) = [0] >= [0] = tt() [0] [0] [1 0 1] [0 0 1] a__U51(tt(),V2) = [0 0 0]V2 >= [0 0 0]V2 = a__U52(a__isNatList(V2)) [0 0 0] [0 0 0] [1] [1] a__U52(tt()) = [0] >= [0] = tt() [0] [0] [1] [1] a__U61(tt(),V2) = [0] >= [0] = a__U62(a__isNatIList(V2)) [0] [0] [1] [1] a__U62(tt()) = [0] >= [0] = tt() [0] [0] [1 0 0] [1] [1 0 0] [1] a__U71(tt(),L,N) = [0 0 0]L + [0] >= [0 0 0]L + [0] = a__U72(a__isNat(N),L) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] a__U72(tt(),L) = [0 0 0]L + [0] >= [0 0 0]L + [0] = s(a__length(mark(L))) [0 0 0] [0] [0 0 0] [0] [1] [1] a__U81(tt()) = [0] >= [0] = nil() [1] [1] [1 0 0] [2] [1 0 0] [2] a__U91(tt(),IL,M,N) = [0 0 0]M + [0] >= [0 0 0]M + [0] = a__U92(a__isNat(M),IL,M,N) [1 0 0] [2] [1 0 0] [2] [1 0 0] [2] [1 0 0] [2] a__U92(tt(),IL,M,N) = [0 0 0]M + [0] >= [0 0 0]M + [0] = a__U93(a__isNat(N),IL,M,N) [1 0 0] [2] [1 0 0] [2] [1 0 0] [2] [1 0 0] [2] a__U93(tt(),IL,M,N) = [0 0 0]M + [0] >= [0 0 0]M + [0] = cons(mark(N),take(M,IL)) [1 0 0] [2] [1 0 0] [2] [1] [1] a__isNat(0()) = [0] >= [0] = tt() [0] [0] [1] [1] a__isNat(length(V1)) = [0] >= [0] = a__U11(a__isNatList(V1)) [0] [0] [1] [1] a__isNat(s(V1)) = [0] >= [0] = a__U21(a__isNat(V1)) [0] [0] [1] [1] a__isNatIList(V) = [0] >= [0] = a__U31(a__isNatList(V)) [0] [0] [1] [1] a__isNatIList(zeros()) = [0] >= [0] = tt() [0] [0] [1] [1] a__isNatIList(cons(V1,V2)) = [0] >= [0] = a__U41(a__isNat(V1),V2) [0] [0] [1] [1] a__isNatList(nil()) = [0] >= [0] = tt() [0] [0] [1 0 1] [1 0 1] a__isNatList(cons(V1,V2)) = [0 0 0]V2 >= [0 0 0]V2 = a__U51(a__isNat(V1),V2) [0 0 0] [0 0 0] [1] [1] a__isNatList(take(V1,V2)) = [0] >= [0] = a__U61(a__isNat(V1),V2) [0] [0] [1] [0] a__length(nil()) = [0] >= [0] = 0() [0] [0] [1 0 1] [1 0 1] a__length(cons(N,L)) = [0 0 0]L >= [0 0 0]L = a__U71(a__isNatList(L),L,N) [0 0 0] [0 0 0] [1] [1] a__take(0(),IL) = [0] >= [0] = a__U81(a__isNatIList(IL)) [1] [1] [1 1 0] [2] [1 0 0] [2] a__take(s(M),cons(N,IL)) = [0 0 0]M + [0] >= [0 0 0]M + [0] = a__U91(a__isNatIList(IL),IL,M,N) [1 1 0] [2] [1 0 0] [2] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [1] [1] mark(U11(X)) = [0] >= [0] = a__U11(mark(X)) [1] [0] [1] [1] mark(U21(X)) = [0] >= [0] = a__U21(mark(X)) [1] [0] [1] [1] mark(U31(X)) = [0] >= [0] = a__U31(mark(X)) [1] [0] [1 0 0] [1 0 0] mark(U41(X1,X2)) = [0 0 0]X1 >= [0 0 0]X1 = a__U41(mark(X1),X2) [1 0 0] [0 0 0] [1] [1] mark(U42(X)) = [0] >= [0] = a__U42(mark(X)) [1] [0] [1] [1] mark(isNatIList(X)) = [0] >= [0] = a__isNatIList(X) [1] [0] [1 0 1] [1 0 1] mark(U51(X1,X2)) = [0 0 0]X2 >= [0 0 0]X2 = a__U51(mark(X1),X2) [1 0 1] [0 0 0] [1 0 0] [1 0 0] mark(U52(X)) = [0 0 0]X >= [0 0 0]X = a__U52(mark(X)) [1 0 0] [0 0 0] [0 0 1] [0 0 1] mark(isNatList(X)) = [0 0 0]X >= [0 0 0]X = a__isNatList(X) [0 0 1] [0 0 0] [1 0 0] [1 0 0] mark(U61(X1,X2)) = [0 0 0]X1 >= [0 0 0]X1 = a__U61(mark(X1),X2) [1 0 0] [0 0 0] [1 0 0] [1 0 0] mark(U62(X)) = [0 0 0]X >= [0 0 0]X = a__U62(mark(X)) [1 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] mark(U71(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = a__U71(mark(X1),X2,X3) [1 1 0] [1 0 0] [1 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] mark(U72(X1,X2)) = [0 0 0]X2 + [0] >= [0 0 0]X2 + [0] = a__U72(mark(X1),X2) [1 0 0] [1] [0 0 0] [0] [1] [1] mark(isNat(X)) = [0] >= [0] = a__isNat(X) [1] [0] [1 0 0] [1 0 0] mark(length(X)) = [0 0 0]X >= [0 0 0]X = a__length(mark(X)) [1 0 0] [0 0 0] [1 0 0] [1 0 0] mark(U81(X)) = [0 0 0]X >= [0 0 0]X = a__U81(mark(X)) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] mark(U91(X1,X2,X3,X4)) = [0 0 0]X1 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X3 + [0] = a__U91(mark(X1),X2,X3,X4) [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] mark(U92(X1,X2,X3,X4)) = [0 0 0]X1 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X3 + [0] = a__U92(mark(X1),X2,X3,X4) [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] mark(U93(X1,X2,X3,X4)) = [0 0 0]X1 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X3 + [0] = a__U93(mark(X1),X2,X3,X4) [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1] [1 0 0] [1] mark(take(X1,X2)) = [0 0 0]X1 + [0] >= [0 0 0]X1 + [0] = a__take(mark(X1),mark(X2)) [1 0 0] [1] [1 0 0] [1] [1 0 1] [1 0 1] mark(cons(X1,X2)) = [0 0 0]X2 >= [0 0 0]X2 = cons(mark(X1),X2) [1 0 1] [1 0 1] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1] [1] mark(tt()) = [0] >= [0] = tt() [1] [0] [1 1 0] [1] [1 0 0] [1] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(mark(X)) [1 1 0] [1] [0 0 0] [0] [1] [1] mark(nil()) = [0] >= [0] = nil() [1] [1] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1] [1] a__U11(X) = [0] >= [0] = U11(X) [0] [0] [1] [1] a__U21(X) = [0] >= [0] = U21(X) [0] [0] [1] [1] a__U31(X) = [0] >= [0] = U31(X) [0] [0] [1 0 0] [1 0 0] a__U41(X1,X2) = [0 0 0]X1 >= [0 0 0]X1 = U41(X1,X2) [0 0 0] [0 0 0] [1] [1] a__U42(X) = [0] >= [0] = U42(X) [0] [0] [1] [1] a__isNatIList(X) = [0] >= [0] = isNatIList(X) [0] [0] [1 0 1] [1 0 1] a__U51(X1,X2) = [0 0 0]X2 >= [0 0 0]X2 = U51(X1,X2) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U52(X) = [0 0 0]X >= [0 0 0]X = U52(X) [0 0 0] [0 0 0] [0 0 1] [0 0 1] a__isNatList(X) = [0 0 0]X >= [0 0 0]X = isNatList(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U61(X1,X2) = [0 0 0]X1 >= [0 0 0]X1 = U61(X1,X2) [0 1 0] [0 0 0] [1 0 0] [1 0 0] a__U62(X) = [0 0 0]X >= [0 0 0]X = U62(X) [0 0 1] [0 0 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] a__U71(X1,X2,X3) = [0 1 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = U71(X1,X2,X3) [0 0 1] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] [1] a__U72(X1,X2) = [0 0 0]X2 + [0] >= [0 0 0]X2 + [0] = U72(X1,X2) [0 0 0] [0] [0 0 0] [0] [1] [1] a__isNat(X) = [0] >= [0] = isNat(X) [0] [0] [1 0 0] [1 0 0] a__length(X) = [0 0 0]X >= [0 0 0]X = length(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U81(X) = [0 0 0]X >= [0 0 0]X = U81(X) [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] a__U91(X1,X2,X3,X4) = [0 0 0]X1 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X3 + [0] = U91(X1,X2,X3,X4) [1 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] a__U92(X1,X2,X3,X4) = [0 0 0]X1 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X3 + [0] = U92(X1,X2,X3,X4) [1 0 0] [1 0 0] [1] [1 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] a__U93(X1,X2,X3,X4) = [0 0 0]X1 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X3 + [0] = U93(X1,X2,X3,X4) [1 0 0] [1 0 0] [1] [1 0 0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] a__take(X1,X2) = [0 0 0]X1 + [0] >= [0 0 0]X1 + [0] = take(X1,X2) [1 0 0] [1] [0 0 0] [1] problem: DPs: a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Restore Modifier: DPs: a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 3/4 DPs: a__isNat#(length(V1)) -> a__isNatList#(V1) a__isNatList#(cons(V1,V2)) -> a__isNat#(V1) a__isNat#(s(V1)) -> a__isNat#(V1) a__isNatList#(cons(V1,V2)) -> a__U51#(a__isNat(V1),V2) a__U51#(tt(),V2) -> a__isNatList#(V2) a__isNatList#(take(V1,V2)) -> a__isNat#(V1) a__isNatList#(take(V1,V2)) -> a__U61#(a__isNat(V1),V2) a__U61#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatList#(V) a__isNatIList#(cons(V1,V2)) -> a__isNat#(V1) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(a__U41#) = 1 pi(a__isNatIList#) = 0 pi(a__U51#) = 1 pi(a__isNatList#) = 0 pi(a__U61#) = 1 pi(a__isNat#) = 0 problem: DPs: a__U51#(tt(),V2) -> a__isNatList#(V2) a__U61#(tt(),V2) -> a__isNatIList#(V2) a__isNatIList#(V) -> a__isNatList#(V) 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(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,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)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) 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) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) SCC Processor: #sccs: 0 #rules: 0 #arcs: 29/16