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=1 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__U72](x0, x1) = 3x0 + 2x1 + 3, [tt] = 0, [a__U62](x0) = 2x0, [a__U41](x0, x1) = x0, [0] = 0, [cons](x0, x1) = x0 + x1, [a__U71](x0, x1, x2) = 3x0 + 2x1 + 3, [s](x0) = x0, [U51](x0, x1) = x0, [a__U93#](x0, x1, x2, x3) = 3/2x0 + 1/2x1 + 1/2x3, [a__U93](x0, x1, x2, x3) = 3x0 + 2x1 + x2 + x3, [mark#](x0) = 1/2x0, [mark](x0) = x0, [U31](x0) = 2x0, [a__isNat](x0) = 0, [a__U71#](x0, x1, x2) = 3/2x0 + x1 + 3/2, [take](x0, x1) = x0 + 2x1, [U42](x0) = 2x0, [a__U81](x0) = 2x0, [length](x0) = 2x0 + 3, [U81](x0) = 2x0, [a__U11](x0) = 2x0, [a__length#](x0) = x0 + 3/2, [U41](x0, x1) = x0, [a__U42](x0) = 2x0, [a__zeros] = 0, [U61](x0, x1) = 2x0, [a__U31](x0) = 2x0, [a__length](x0) = 2x0 + 3, [a__U92](x0, x1, x2, x3) = 3x0 + 2x1 + x2 + x3, [a__U72#](x0, x1) = 3/2x0 + x1 + 3/2, [U11](x0) = 2x0, [a__U61](x0, x1) = 2x0, [U21](x0) = x0, [a__isNatList](x0) = 0, [a__take#](x0, x1) = 1/2x0 + 1/2x1, [U91](x0, x1, x2, x3) = 3x0 + 2x1 + x2 + x3, [nil] = 0, [a__U91](x0, x1, x2, x3) = 3x0 + 2x1 + x2 + x3, [a__U51](x0, x1) = x0, [a__isNatIList](x0) = 0, [a__U52](x0) = 2x0, [U62](x0) = 2x0, [a__U92#](x0, x1, x2, x3) = 3/2x0 + 1/2x1 + 1/2x2 + 1/2x3, [a__U21](x0) = x0, [isNatList](x0) = 0, [U93](x0, x1, x2, x3) = 3x0 + 2x1 + x2 + x3, [U52](x0) = 2x0, [U71](x0, x1, x2) = 3x0 + 2x1 + 3, [U72](x0, x1) = 3x0 + 2x1 + 3, [U92](x0, x1, x2, x3) = 3x0 + 2x1 + x2 + x3, [a__U91#](x0, x1, x2, x3) = 3/2x0 + 1/2x1 + 1/2x2 + 1/2x3, [isNatIList](x0) = 0, [a__take](x0, x1) = x0 + 2x1, [isNat](x0) = 0, [zeros] = 0 orientation: a__take#(s(M),cons(N,IL)) = 1/2IL + 1/2M + 1/2N >= 1/2IL + 1/2M + 1/2N = a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) = 1/2IL + 1/2M + 1/2N >= 1/2IL + 1/2M + 1/2N = a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) = 1/2IL + 1/2M + 1/2N >= 1/2IL + 1/2N = a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) = 1/2IL + 1/2N >= 1/2N = mark#(N) mark#(U11(X)) = X >= 1/2X = mark#(X) mark#(U21(X)) = 1/2X >= 1/2X = mark#(X) mark#(U31(X)) = X >= 1/2X = mark#(X) mark#(U41(X1,X2)) = 1/2X1 >= 1/2X1 = mark#(X1) mark#(U42(X)) = X >= 1/2X = mark#(X) mark#(U51(X1,X2)) = 1/2X1 >= 1/2X1 = mark#(X1) mark#(U52(X)) = X >= 1/2X = mark#(X) mark#(U61(X1,X2)) = X1 >= 1/2X1 = mark#(X1) mark#(U62(X)) = X >= 1/2X = mark#(X) mark#(U71(X1,X2,X3)) = 3/2X1 + X2 + 3/2 >= 1/2X1 = mark#(X1) mark#(U71(X1,X2,X3)) = 3/2X1 + X2 + 3/2 >= 3/2X1 + X2 + 3/2 = a__U71#(mark(X1),X2,X3) a__U71#(tt(),L,N) = L + 3/2 >= L + 3/2 = a__U72#(a__isNat(N),L) a__U72#(tt(),L) = L + 3/2 >= 1/2L = mark#(L) mark#(U72(X1,X2)) = 3/2X1 + X2 + 3/2 >= 1/2X1 = mark#(X1) mark#(U72(X1,X2)) = 3/2X1 + X2 + 3/2 >= 3/2X1 + X2 + 3/2 = a__U72#(mark(X1),X2) a__U72#(tt(),L) = L + 3/2 >= L + 3/2 = a__length#(mark(L)) a__length#(cons(N,L)) = L + N + 3/2 >= L + 3/2 = a__U71#(a__isNatList(L),L,N) mark#(length(X)) = X + 3/2 >= 1/2X = mark#(X) mark#(length(X)) = X + 3/2 >= X + 3/2 = a__length#(mark(X)) mark#(U81(X)) = X >= 1/2X = mark#(X) mark#(U91(X1,X2,X3,X4)) = 3/2X1 + X2 + 1/2X3 + 1/2X4 >= 1/2X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = 3/2X1 + X2 + 1/2X3 + 1/2X4 >= 3/2X1 + 1/2X2 + 1/2X3 + 1/2X4 = a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) = 3/2X1 + X2 + 1/2X3 + 1/2X4 >= 1/2X1 = mark#(X1) mark#(U92(X1,X2,X3,X4)) = 3/2X1 + X2 + 1/2X3 + 1/2X4 >= 3/2X1 + 1/2X2 + 1/2X3 + 1/2X4 = a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) = 3/2X1 + X2 + 1/2X3 + 1/2X4 >= 1/2X1 = mark#(X1) mark#(U93(X1,X2,X3,X4)) = 3/2X1 + X2 + 1/2X3 + 1/2X4 >= 3/2X1 + 1/2X2 + 1/2X4 = a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) = 1/2X1 + X2 >= 1/2X2 = mark#(X2) mark#(take(X1,X2)) = 1/2X1 + X2 >= 1/2X1 = mark#(X1) mark#(take(X1,X2)) = 1/2X1 + X2 >= 1/2X1 + 1/2X2 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = 1/2X1 + 1/2X2 >= 1/2X1 = mark#(X1) mark#(s(X)) = 1/2X >= 1/2X = 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(),V2) = 0 >= 0 = a__U62(a__isNatIList(V2)) a__U62(tt()) = 0 >= 0 = tt() a__U71(tt(),L,N) = 2L + 3 >= 2L + 3 = a__U72(a__isNat(N),L) a__U72(tt(),L) = 2L + 3 >= 2L + 3 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 2IL + M + N >= 2IL + M + N = a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) = 2IL + M + N >= 2IL + M + N = a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) = 2IL + M + N >= 2IL + M + N = cons(mark(N),take(M,IL)) 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__isNatList(take(V1,V2)) = 0 >= 0 = a__U61(a__isNat(V1),V2) a__length(nil()) = 3 >= 0 = 0() a__length(cons(N,L)) = 2L + 2N + 3 >= 2L + 3 = a__U71(a__isNatList(L),L,N) a__take(0(),IL) = 2IL >= 0 = a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) = 2IL + M + 2N >= 2IL + M + N = a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X)) = 2X >= 2X = a__U11(mark(X)) mark(U21(X)) = X >= X = a__U21(mark(X)) mark(U31(X)) = 2X >= 2X = a__U31(mark(X)) mark(U41(X1,X2)) = X1 >= X1 = a__U41(mark(X1),X2) mark(U42(X)) = 2X >= 2X = 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)) = 2X >= 2X = a__U52(mark(X)) mark(isNatList(X)) = 0 >= 0 = a__isNatList(X) mark(U61(X1,X2)) = 2X1 >= 2X1 = a__U61(mark(X1),X2) mark(U62(X)) = 2X >= 2X = a__U62(mark(X)) mark(U71(X1,X2,X3)) = 3X1 + 2X2 + 3 >= 3X1 + 2X2 + 3 = a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) = 3X1 + 2X2 + 3 >= 3X1 + 2X2 + 3 = a__U72(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = 2X + 3 >= 2X + 3 = a__length(mark(X)) mark(U81(X)) = 2X >= 2X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 3X1 + 2X2 + X3 + X4 >= 3X1 + 2X2 + X3 + X4 = a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) = 3X1 + 2X2 + X3 + X4 >= 3X1 + 2X2 + X3 + X4 = a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) = 3X1 + 2X2 + X3 + X4 >= 3X1 + 2X2 + X3 + X4 = a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + 2X2 >= X1 + 2X2 = a__take(mark(X1),mark(X2)) 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) = 2X >= 2X = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = 2X >= 2X = U31(X) a__U41(X1,X2) = X1 >= X1 = U41(X1,X2) a__U42(X) = 2X >= 2X = U42(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2) = X1 >= X1 = U51(X1,X2) a__U52(X) = 2X >= 2X = U52(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U61(X1,X2) = 2X1 >= 2X1 = U61(X1,X2) a__U62(X) = 2X >= 2X = U62(X) a__U71(X1,X2,X3) = 3X1 + 2X2 + 3 >= 3X1 + 2X2 + 3 = U71(X1,X2,X3) a__U72(X1,X2) = 3X1 + 2X2 + 3 >= 3X1 + 2X2 + 3 = U72(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 2X + 3 >= 2X + 3 = length(X) a__U81(X) = 2X >= 2X = U81(X) a__U91(X1,X2,X3,X4) = 3X1 + 2X2 + X3 + X4 >= 3X1 + 2X2 + X3 + X4 = U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) = 3X1 + 2X2 + X3 + X4 >= 3X1 + 2X2 + X3 + X4 = U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) = 3X1 + 2X2 + X3 + X4 >= 3X1 + 2X2 + X3 + X4 = U93(X1,X2,X3,X4) a__take(X1,X2) = X1 + 2X2 >= X1 + 2X2 = take(X1,X2) problem: 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)) -> a__U71#(mark(X1),X2,X3) a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) 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)) -> 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) Restore Modifier: 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)) -> a__U71#(mark(X1),X2,X3) a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) 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)) -> 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) SCC Processor: #sccs: 2 #rules: 28 #arcs: 609/961 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#(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) Arctic Interpretation Processor: dimension: 1 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__U72](x0, x1) = 1x0 + x1 + 0, [tt] = 0, [a__U62](x0) = x0 + 0, [a__U41](x0, x1) = x0 + x1 + 0, [0] = 0, [cons](x0, x1) = 1x0 + x1 + 0, [a__U71](x0, x1, x2) = x0 + x1 + 1x2 + 0, [s](x0) = x0 + 0, [U51](x0, x1) = x0 + x1, [a__U93#](x0, x1, x2, x3) = x0 + 1x3, [a__U93](x0, x1, x2, x3) = 1x0 + x1 + x2 + 1x3, [mark#](x0) = x0, [mark](x0) = x0, [U31](x0) = x0, [a__isNat](x0) = x0, [take](x0, x1) = x0 + x1, [U42](x0) = x0, [a__U81](x0) = x0, [length](x0) = x0 + 0, [U81](x0) = x0, [a__U11](x0) = x0 + 0, [U41](x0, x1) = x0 + x1 + 0, [a__U42](x0) = x0, [a__zeros] = 1, [U61](x0, x1) = x0 + x1 + 0, [a__U31](x0) = x0, [a__length](x0) = x0 + 0, [a__U92](x0, x1, x2, x3) = x0 + x1 + x2 + 1x3 + 0, [U11](x0) = x0 + 0, [a__U61](x0, x1) = x0 + x1 + 0, [U21](x0) = x0, [a__isNatList](x0) = x0 + 0, [a__take#](x0, x1) = x0 + x1, [U91](x0, x1, x2, x3) = x0 + x1 + x2 + 1x3 + 0, [nil] = 0, [a__U91](x0, x1, x2, x3) = x0 + x1 + x2 + 1x3 + 0, [a__U51](x0, x1) = x0 + x1, [a__isNatIList](x0) = x0 + 0, [a__U52](x0) = x0 + 0, [U62](x0) = x0 + 0, [a__U92#](x0, x1, x2, x3) = x0 + x2 + 1x3, [a__U21](x0) = x0, [isNatList](x0) = x0 + 0, [U93](x0, x1, x2, x3) = 1x0 + x1 + x2 + 1x3, [U52](x0) = x0 + 0, [U71](x0, x1, x2) = x0 + x1 + 1x2 + 0, [U72](x0, x1) = 1x0 + x1 + 0, [U92](x0, x1, x2, x3) = x0 + x1 + x2 + 1x3 + 0, [a__U91#](x0, x1, x2, x3) = x1 + x2 + 1x3 + 0, [isNatIList](x0) = x0 + 0, [a__take](x0, x1) = x0 + x1, [isNat](x0) = x0, [zeros] = 1 orientation: a__take#(s(M),cons(N,IL)) = IL + M + 1N + 0 >= IL + M + 1N + 0 = a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) = IL + M + 1N + 0 >= M + 1N = a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) = M + 1N + 0 >= 1N = a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) = 1N + 0 >= N = mark#(N) mark#(U11(X)) = X + 0 >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) mark#(U31(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(U42(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U52(X)) = X + 0 >= X = mark#(X) mark#(U61(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(U62(X)) = X + 0 >= X = mark#(X) mark#(U81(X)) = X >= X = mark#(X) mark#(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + 1X4 + 0 >= X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + 1X4 + 0 >= X2 + X3 + 1X4 + 0 = a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) = X1 + X2 + X3 + 1X4 + 0 >= X1 = mark#(X1) mark#(U92(X1,X2,X3,X4)) = X1 + X2 + X3 + 1X4 + 0 >= X1 + X3 + 1X4 = a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) = 1X1 + X2 + X3 + 1X4 >= X1 = mark#(X1) mark#(U93(X1,X2,X3,X4)) = 1X1 + X2 + X3 + 1X4 >= X1 + 1X4 = a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = 1X1 + X2 + 0 >= X1 = mark#(X1) mark#(s(X)) = X + 0 >= 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 + 0 >= V2 + 0 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 0 >= 0 = tt() a__U51(tt(),V2) = V2 + 0 >= V2 + 0 = a__U52(a__isNatList(V2)) a__U52(tt()) = 0 >= 0 = tt() a__U61(tt(),V2) = V2 + 0 >= V2 + 0 = a__U62(a__isNatIList(V2)) a__U62(tt()) = 0 >= 0 = tt() a__U71(tt(),L,N) = L + 1N + 0 >= L + 1N + 0 = a__U72(a__isNat(N),L) a__U72(tt(),L) = L + 1 >= L + 0 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + M + 1N + 0 >= IL + M + 1N + 0 = a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) = IL + M + 1N + 0 >= IL + M + 1N = a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) = IL + M + 1N + 1 >= IL + M + 1N + 0 = cons(mark(N),take(M,IL)) a__isNat(0()) = 0 >= 0 = tt() a__isNat(length(V1)) = V1 + 0 >= V1 + 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = V1 + 0 >= V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = V + 0 >= V + 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 1 >= 0 = tt() a__isNatIList(cons(V1,V2)) = 1V1 + V2 + 0 >= V1 + V2 + 0 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = 1V1 + V2 + 0 >= V1 + V2 = a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) = V1 + V2 + 0 >= V1 + V2 + 0 = a__U61(a__isNat(V1),V2) a__length(nil()) = 0 >= 0 = 0() a__length(cons(N,L)) = L + 1N + 0 >= L + 1N + 0 = a__U71(a__isNatList(L),L,N) a__take(0(),IL) = IL + 0 >= IL + 0 = a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) = IL + M + 1N + 0 >= IL + M + 1N + 0 = a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) = 1 >= 1 = a__zeros() mark(U11(X)) = X + 0 >= X + 0 = 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 + X2 + 0 >= X1 + X2 + 0 = a__U41(mark(X1),X2) mark(U42(X)) = X >= X = a__U42(mark(X)) mark(isNatIList(X)) = X + 0 >= X + 0 = a__isNatIList(X) mark(U51(X1,X2)) = X1 + X2 >= X1 + X2 = a__U51(mark(X1),X2) mark(U52(X)) = X + 0 >= X + 0 = a__U52(mark(X)) mark(isNatList(X)) = X + 0 >= X + 0 = a__isNatList(X) mark(U61(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U61(mark(X1),X2) mark(U62(X)) = X + 0 >= X + 0 = a__U62(mark(X)) mark(U71(X1,X2,X3)) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) = 1X1 + X2 + 0 >= 1X1 + X2 + 0 = a__U72(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(length(X)) = X + 0 >= X + 0 = a__length(mark(X)) mark(U81(X)) = X >= X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + 1X4 + 0 >= X1 + X2 + X3 + 1X4 + 0 = a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) = X1 + X2 + X3 + 1X4 + 0 >= X1 + X2 + X3 + 1X4 + 0 = a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) = 1X1 + X2 + X3 + 1X4 >= 1X1 + X2 + X3 + 1X4 = a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = 1X1 + X2 + 0 >= 1X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 1 >= 1 = zeros() a__U11(X) = X + 0 >= X + 0 = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = X >= X = U31(X) a__U41(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = X + 0 >= X + 0 = isNatIList(X) a__U51(X1,X2) = X1 + X2 >= X1 + X2 = U51(X1,X2) a__U52(X) = X + 0 >= X + 0 = U52(X) a__isNatList(X) = X + 0 >= X + 0 = isNatList(X) a__U61(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U61(X1,X2) a__U62(X) = X + 0 >= X + 0 = U62(X) a__U71(X1,X2,X3) = X1 + X2 + 1X3 + 0 >= X1 + X2 + 1X3 + 0 = U71(X1,X2,X3) a__U72(X1,X2) = 1X1 + X2 + 0 >= 1X1 + X2 + 0 = U72(X1,X2) a__isNat(X) = X >= X = isNat(X) a__length(X) = X + 0 >= X + 0 = length(X) a__U81(X) = X >= X = U81(X) a__U91(X1,X2,X3,X4) = X1 + X2 + X3 + 1X4 + 0 >= X1 + X2 + X3 + 1X4 + 0 = U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) = X1 + X2 + X3 + 1X4 + 0 >= X1 + X2 + X3 + 1X4 + 0 = U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) = 1X1 + X2 + X3 + 1X4 >= 1X1 + X2 + X3 + 1X4 = U93(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) problem: 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) 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#(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)) -> 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#(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__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) 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#(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)) -> 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#(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: 1 #rules: 15 #arcs: 385/484 DPs: mark#(U92(X1,X2,X3,X4)) -> mark#(X1) 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#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) mark#(take(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 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(),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) Qed 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=1 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__U72](x0, x1) = 3/2x1, [tt] = 1, [a__U62](x0) = 1, [a__U41](x0, x1) = 2, [0] = 1/2, [cons](x0, x1) = 3/2x1, [a__U71](x0, x1, x2) = 3/2x1, [s](x0) = 3/2x0, [U51](x0, x1) = x1, [a__U93](x0, x1, x2, x3) = 3x2, [U31](x0) = 2, [mark](x0) = x0, [a__isNat](x0) = 2x0, [a__U71#](x0, x1, x2) = x0 + 2x1 + 5/2, [take](x0, x1) = 2x0, [U42](x0) = x0, [a__U81](x0) = 1, [length](x0) = x0, [U81](x0) = 1, [a__U11](x0) = x0, [U41](x0, x1) = 2, [a__length#](x0) = 2x0 + 7/2, [a__U42](x0) = x0, [a__zeros] = 0, [U61](x0, x1) = x0, [a__U31](x0) = 2, [a__length](x0) = x0, [a__U92](x0, x1, x2, x3) = 3x2, [U11](x0) = x0, [a__U72#](x0, x1) = 2x1 + 7/2, [a__U61](x0, x1) = x0, [U21](x0) = x0, [a__isNatList](x0) = x0, [U91](x0, x1, x2, x3) = 3x2, [nil] = 1, [a__U91](x0, x1, x2, x3) = 3x2, [a__U51](x0, x1) = x1, [a__isNatIList](x0) = 2, [a__U52](x0) = x0, [U62](x0) = 1, [a__U21](x0) = x0, [isNatList](x0) = x0, [U93](x0, x1, x2, x3) = 3x2, [U52](x0) = x0, [U71](x0, x1, x2) = 3/2x1, [U72](x0, x1) = 3/2x1, [U92](x0, x1, x2, x3) = 3x2, [isNatIList](x0) = 2, [a__take](x0, x1) = 2x0, [isNat](x0) = 2x0, [zeros] = 0 orientation: a__U71#(tt(),L,N) = 2L + 7/2 >= 2L + 7/2 = a__U72#(a__isNat(N),L) a__U72#(tt(),L) = 2L + 7/2 >= 2L + 7/2 = a__length#(mark(L)) a__length#(cons(N,L)) = 3L + 7/2 >= 3L + 5/2 = a__U71#(a__isNatList(L),L,N) a__zeros() = 0 >= 0 = cons(0(),zeros()) a__U11(tt()) = 1 >= 1 = tt() a__U21(tt()) = 1 >= 1 = tt() a__U31(tt()) = 2 >= 1 = tt() a__U41(tt(),V2) = 2 >= 2 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 1 >= 1 = tt() a__U51(tt(),V2) = V2 >= V2 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 1 = tt() a__U61(tt(),V2) = 1 >= 1 = a__U62(a__isNatIList(V2)) a__U62(tt()) = 1 >= 1 = tt() a__U71(tt(),L,N) = 3/2L >= 3/2L = a__U72(a__isNat(N),L) a__U72(tt(),L) = 3/2L >= 3/2L = s(a__length(mark(L))) a__U81(tt()) = 1 >= 1 = nil() a__U91(tt(),IL,M,N) = 3M >= 3M = a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) = 3M >= 3M = a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) = 3M >= 3M = cons(mark(N),take(M,IL)) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 2V1 >= V1 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = 3V1 >= 2V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = 2 >= 2 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 2 >= 1 = tt() a__isNatIList(cons(V1,V2)) = 2 >= 2 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = 3/2V2 >= V2 = a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) = 2V1 >= 2V1 = a__U61(a__isNat(V1),V2) a__length(nil()) = 1 >= 1/2 = 0() a__length(cons(N,L)) = 3/2L >= 3/2L = a__U71(a__isNatList(L),L,N) a__take(0(),IL) = 1 >= 1 = a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) = 3M >= 3M = a__U91(a__isNatIList(IL),IL,M,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)) = 2 >= 2 = a__U31(mark(X)) mark(U41(X1,X2)) = 2 >= 2 = a__U41(mark(X1),X2) mark(U42(X)) = X >= X = a__U42(mark(X)) mark(isNatIList(X)) = 2 >= 2 = a__isNatIList(X) mark(U51(X1,X2)) = X2 >= 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)) = X1 >= X1 = a__U61(mark(X1),X2) mark(U62(X)) = 1 >= 1 = a__U62(mark(X)) mark(U71(X1,X2,X3)) = 3/2X2 >= 3/2X2 = a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) = 3/2X2 >= 3/2X2 = a__U72(mark(X1),X2) mark(isNat(X)) = 2X >= 2X = a__isNat(X) mark(length(X)) = X >= X = a__length(mark(X)) mark(U81(X)) = 1 >= 1 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 3X3 >= 3X3 = a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) = 3X3 >= 3X3 = a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) = 3X3 >= 3X3 = a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 2X1 >= 2X1 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = 3/2X2 >= 3/2X2 = cons(mark(X1),X2) mark(0()) = 1/2 >= 1/2 = 0() mark(tt()) = 1 >= 1 = tt() mark(s(X)) = 3/2X >= 3/2X = 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) = 2 >= 2 = U31(X) a__U41(X1,X2) = 2 >= 2 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 2 >= 2 = isNatIList(X) a__U51(X1,X2) = X2 >= X2 = U51(X1,X2) a__U52(X) = X >= X = U52(X) a__isNatList(X) = X >= X = isNatList(X) a__U61(X1,X2) = X1 >= X1 = U61(X1,X2) a__U62(X) = 1 >= 1 = U62(X) a__U71(X1,X2,X3) = 3/2X2 >= 3/2X2 = U71(X1,X2,X3) a__U72(X1,X2) = 3/2X2 >= 3/2X2 = U72(X1,X2) a__isNat(X) = 2X >= 2X = isNat(X) a__length(X) = X >= X = length(X) a__U81(X) = 1 >= 1 = U81(X) a__U91(X1,X2,X3,X4) = 3X3 >= 3X3 = U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) = 3X3 >= 3X3 = U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) = 3X3 >= 3X3 = U93(X1,X2,X3,X4) a__take(X1,X2) = 2X1 >= 2X1 = take(X1,X2) problem: DPs: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) a__U72#(tt(),L) -> a__length#(mark(L)) 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__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) a__U72#(tt(),L) -> a__length#(mark(L)) 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