MAYBE Problem: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),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) Open 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