/export/starexec/sandbox/solver/bin/starexec_run_ttt2 /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- failure 'Failure("No result of SMT solver - maybe due to some flag of the solver? or the solver was not found in PATH?")' in subprocess 9921 failure 'Failure("No result of SMT solver - maybe due to some flag of the solver? or the solver was not found in PATH?")' in subprocess 9980 failure 'Failure("No result of SMT solver - maybe due to some flag of the solver? or the solver was not found in PATH?")' in subprocess 10036 failure 'Failure("No result of SMT solver - maybe due to some flag of the solver? or the solver was not found in PATH?")' in subprocess 10089 failure 'Failure("No result of SMT solver - maybe due to some flag of the solver? or the solver was not found in PATH?")' in subprocess 10138 failure 'Failure("No result of SMT solver - maybe due to some flag of the solver? or the solver was not found in PATH?")' in subprocess 10242 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) 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) = x0 + 3x1 + 3, [tt] = 0, [a__U62](x0) = x0, [a__U41](x0, x1) = x0 + x1 + 0, [0] = 3, [cons](x0, x1) = x0 + x1 + 0, [a__U71](x0, x1, x2) = 3x0 + 3x1 + x2 + 3, [s](x0) = x0, [U51](x0, x1) = x0 + x1 + 0, [a__U93#](x0, x1, x2, x3) = x1 + x3, [a__U93](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 4, [mark#](x0) = x0, [mark](x0) = x0 + 0, [U31](x0) = x0, [a__isNat](x0) = x0 + 0, [a__U71#](x0, x1, x2) = 3x0 + 3x1, [take](x0, x1) = x0 + x1 + 4, [U42](x0) = x0, [a__U81](x0) = x0 + 0, [length](x0) = 3x0 + 3, [U81](x0) = x0, [a__U11](x0) = x0, [a__length#](x0) = 3x0 + 0, [U41](x0, x1) = x0 + x1, [a__U42](x0) = x0 + 0, [a__zeros] = 3, [U61](x0, x1) = x0 + x1, [a__U31](x0) = x0, [a__length](x0) = 3x0 + 3, [a__U92](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 4, [a__U72#](x0, x1) = 3x1 + 3, [U11](x0) = x0, [a__U61](x0, x1) = x0 + x1 + 0, [U21](x0) = x0, [a__isNatList](x0) = x0, [a__take#](x0, x1) = x0 + x1 + 0, [U91](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 4, [nil] = 0, [a__U91](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 4, [a__U51](x0, x1) = x0 + x1 + 0, [a__isNatIList](x0) = x0, [a__U52](x0) = x0 + 0, [U62](x0) = x0, [a__U92#](x0, x1, x2, x3) = x0 + x1 + x3 + 0, [a__U21](x0) = x0 + 0, [isNatList](x0) = x0, [U93](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 4, [U52](x0) = x0, [U71](x0, x1, x2) = 3x0 + 3x1 + x2 + 3, [U72](x0, x1) = x0 + 3x1 + 3, [U92](x0, x1, x2, x3) = x0 + x1 + x2 + x3 + 4, [a__U91#](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [isNatIList](x0) = x0, [a__take](x0, x1) = x0 + x1 + 4, [isNat](x0) = x0 + 0, [zeros] = 3 orientation: a__take#(s(M),cons(N,IL)) = IL + M + N + 0 >= IL + M + N = a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) = IL + M + N + 0 >= IL + M + N + 0 = a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) = IL + N + 0 >= IL + N = a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) = IL + N >= N = mark#(N) mark#(U11(X)) = X >= X = mark#(X) mark#(U21(X)) = X >= X = mark#(X) mark#(U31(X)) = X >= X = mark#(X) mark#(U41(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U42(X)) = X >= X = mark#(X) mark#(U51(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(U52(X)) = X >= X = mark#(X) mark#(U61(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(U62(X)) = X >= X = mark#(X) mark#(U71(X1,X2,X3)) = 3X1 + 3X2 + X3 + 3 >= X1 = mark#(X1) mark#(U71(X1,X2,X3)) = 3X1 + 3X2 + X3 + 3 >= 3X1 + 3X2 + 3 = a__U71#(mark(X1),X2,X3) a__U71#(tt(),L,N) = 3L + 3 >= 3L + 3 = a__U72#(a__isNat(N),L) a__U72#(tt(),L) = 3L + 3 >= L = mark#(L) mark#(U72(X1,X2)) = X1 + 3X2 + 3 >= X1 = mark#(X1) mark#(U72(X1,X2)) = X1 + 3X2 + 3 >= 3X2 + 3 = a__U72#(mark(X1),X2) a__U72#(tt(),L) = 3L + 3 >= 3L + 3 = a__length#(mark(L)) a__length#(cons(N,L)) = 3L + 3N + 3 >= 3L = a__U71#(a__isNatList(L),L,N) mark#(length(X)) = 3X + 3 >= X = mark#(X) mark#(length(X)) = 3X + 3 >= 3X + 3 = a__length#(mark(X)) mark#(U81(X)) = X >= X = mark#(X) mark#(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X1 + X2 + X3 + X4 + 0 = a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X1 = mark#(X1) mark#(U92(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X1 + X2 + X4 + 0 = a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X1 = mark#(X1) mark#(U93(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X2 + X4 = a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) = X1 + X2 + 4 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 4 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 0 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__zeros() = 3 >= 3 = 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 = a__U62(a__isNatIList(V2)) a__U62(tt()) = 0 >= 0 = tt() a__U71(tt(),L,N) = 3L + N + 3 >= 3L + N + 3 = a__U72(a__isNat(N),L) a__U72(tt(),L) = 3L + 3 >= 3L + 3 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = IL + M + N + 4 >= IL + M + N + 4 = a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) = IL + M + N + 4 >= IL + M + N + 4 = a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) = IL + M + N + 4 >= IL + M + N + 4 = cons(mark(N),take(M,IL)) a__isNat(0()) = 3 >= 0 = tt() a__isNat(length(V1)) = 3V1 + 3 >= V1 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = V1 + 0 >= V1 + 0 = a__U21(a__isNat(V1)) a__isNatIList(V) = V >= V = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 3 >= 0 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 + 0 >= V1 + V2 + 0 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 0 >= 0 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 + 0 >= V1 + V2 + 0 = a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) = V1 + V2 + 4 >= V1 + V2 + 0 = a__U61(a__isNat(V1),V2) a__length(nil()) = 3 >= 3 = 0() a__length(cons(N,L)) = 3L + 3N + 3 >= 3L + N + 3 = a__U71(a__isNatList(L),L,N) a__take(0(),IL) = IL + 4 >= IL + 0 = a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) = IL + M + N + 4 >= IL + M + N + 4 = a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) = 3 >= 3 = a__zeros() mark(U11(X)) = X + 0 >= X + 0 = a__U11(mark(X)) mark(U21(X)) = X + 0 >= X + 0 = a__U21(mark(X)) mark(U31(X)) = X + 0 >= X + 0 = a__U31(mark(X)) mark(U41(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U41(mark(X1),X2) mark(U42(X)) = X + 0 >= X + 0 = a__U42(mark(X)) mark(isNatIList(X)) = X + 0 >= X = a__isNatIList(X) mark(U51(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U51(mark(X1),X2) mark(U52(X)) = X + 0 >= X + 0 = a__U52(mark(X)) mark(isNatList(X)) = X + 0 >= X = 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)) = 3X1 + 3X2 + X3 + 3 >= 3X1 + 3X2 + X3 + 3 = a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) = X1 + 3X2 + 3 >= X1 + 3X2 + 3 = a__U72(mark(X1),X2) mark(isNat(X)) = X + 0 >= X + 0 = a__isNat(X) mark(length(X)) = 3X + 3 >= 3X + 3 = a__length(mark(X)) mark(U81(X)) = X + 0 >= X + 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X1 + X2 + X3 + X4 + 4 = a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X1 + X2 + X3 + X4 + 4 = a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) = X1 + X2 + X3 + X4 + 4 >= X1 + X2 + X3 + X4 + 4 = a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 3 >= 3 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 3 >= 3 = zeros() a__U11(X) = X >= X = U11(X) a__U21(X) = X + 0 >= X = U21(X) a__U31(X) = X >= X = U31(X) a__U41(X1,X2) = X1 + X2 + 0 >= X1 + X2 = U41(X1,X2) a__U42(X) = X + 0 >= X = U42(X) a__isNatIList(X) = X >= X = isNatIList(X) a__U51(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U51(X1,X2) a__U52(X) = X + 0 >= X = U52(X) a__isNatList(X) = X >= X = isNatList(X) a__U61(X1,X2) = X1 + X2 + 0 >= X1 + X2 = U61(X1,X2) a__U62(X) = X >= X = U62(X) a__U71(X1,X2,X3) = 3X1 + 3X2 + X3 + 3 >= 3X1 + 3X2 + X3 + 3 = U71(X1,X2,X3) a__U72(X1,X2) = X1 + 3X2 + 3 >= X1 + 3X2 + 3 = U72(X1,X2) a__isNat(X) = X + 0 >= X + 0 = isNat(X) a__length(X) = 3X + 3 >= 3X + 3 = length(X) a__U81(X) = X + 0 >= X = U81(X) a__U91(X1,X2,X3,X4) = X1 + X2 + X3 + X4 + 4 >= X1 + X2 + X3 + X4 + 4 = U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) = X1 + X2 + X3 + X4 + 4 >= X1 + X2 + X3 + X4 + 4 = U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) = X1 + X2 + X3 + X4 + 4 >= X1 + X2 + X3 + X4 + 4 = U93(X1,X2,X3,X4) a__take(X1,X2) = X1 + X2 + 4 >= X1 + X2 + 4 = 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)) -> mark#(X1) mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2) a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) -> mark#(X1) mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) -> mark#(X1) mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> 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)) -> mark#(X1) mark#(U72(X1,X2)) -> a__U72#(mark(X1),X2) a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) mark#(length(X)) -> a__length#(mark(X)) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) -> mark#(X1) mark#(U92(X1,X2,X3,X4)) -> a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) -> mark#(X1) mark#(U93(X1,X2,X3,X4)) -> a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) -> 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: 29 #arcs: 609/1024 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#(U72(X1,X2)) -> mark#(X1) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) 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) = x0 + 4x1, [tt] = 1, [a__U62](x0) = 2x0, [a__U41](x0, x1) = x0 + x1 + 2, [0] = 1, [cons](x0, x1) = x0 + x1 + 2, [a__U71](x0, x1, x2) = 4x1 + x2, [s](x0) = x0, [U51](x0, x1) = x0 + x1, [a__U93#](x0, x1, x2, x3) = x0 + x1 + x2 + x3, [a__U93](x0, x1, x2, x3) = 4x0 + 4x1 + 6x2 + 1x3, [mark#](x0) = x0 + 0, [mark](x0) = x0, [U31](x0) = x0, [a__isNat](x0) = x0, [take](x0, x1) = 6x0 + 4x1 + 2, [U42](x0) = x0 + 0, [a__U81](x0) = 2x0, [length](x0) = 4x0, [U81](x0) = 2x0, [a__U11](x0) = 4x0, [U41](x0, x1) = x0 + x1 + 2, [a__U42](x0) = x0 + 0, [a__zeros] = 2, [U61](x0, x1) = 1x0 + 4x1, [a__U31](x0) = x0, [a__length](x0) = 4x0, [a__U92](x0, x1, x2, x3) = x0 + 4x1 + 6x2 + 4x3 + 1, [U11](x0) = 4x0, [a__U61](x0, x1) = 1x0 + 4x1, [U21](x0) = x0, [a__isNatList](x0) = x0, [a__take#](x0, x1) = 6x0 + 4x1, [U91](x0, x1, x2, x3) = x0 + 4x1 + 6x2 + 4x3 + 0, [nil] = 1, [a__U91](x0, x1, x2, x3) = x0 + 4x1 + 6x2 + 4x3 + 0, [a__U51](x0, x1) = x0 + x1, [a__isNatIList](x0) = x0 + 0, [a__U52](x0) = x0, [U62](x0) = 2x0, [a__U92#](x0, x1, x2, x3) = x1 + 3x2 + 4x3, [a__U21](x0) = x0, [isNatList](x0) = x0, [U93](x0, x1, x2, x3) = 4x0 + 4x1 + 6x2 + 1x3, [U52](x0) = x0, [U71](x0, x1, x2) = 4x1 + x2, [U72](x0, x1) = x0 + 4x1, [U92](x0, x1, x2, x3) = x0 + 4x1 + 6x2 + 4x3 + 1, [a__U91#](x0, x1, x2, x3) = 4x1 + 6x2 + 4x3, [isNatIList](x0) = x0 + 0, [a__take](x0, x1) = 6x0 + 4x1 + 2, [isNat](x0) = x0, [zeros] = 2 orientation: a__take#(s(M),cons(N,IL)) = 4IL + 6M + 4N + 6 >= 4IL + 6M + 4N = a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) = 4IL + 6M + 4N >= IL + 3M + 4N = a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) = IL + 3M + 4N >= IL + M + N = a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) = IL + M + N + 1 >= N + 0 = mark#(N) mark#(U11(X)) = 4X + 0 >= X + 0 = mark#(X) mark#(U21(X)) = X + 0 >= X + 0 = mark#(X) mark#(U31(X)) = X + 0 >= X + 0 = mark#(X) mark#(U41(X1,X2)) = X1 + X2 + 2 >= X1 + 0 = mark#(X1) mark#(U42(X)) = X + 0 >= X + 0 = mark#(X) mark#(U51(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(U52(X)) = X + 0 >= X + 0 = mark#(X) mark#(U61(X1,X2)) = 1X1 + 4X2 + 0 >= X1 + 0 = mark#(X1) mark#(U62(X)) = 2X + 0 >= X + 0 = mark#(X) mark#(U72(X1,X2)) = X1 + 4X2 + 0 >= X1 + 0 = mark#(X1) mark#(U81(X)) = 2X + 0 >= X + 0 = mark#(X) mark#(U91(X1,X2,X3,X4)) = X1 + 4X2 + 6X3 + 4X4 + 0 >= X1 + 0 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + 4X2 + 6X3 + 4X4 + 0 >= 4X2 + 6X3 + 4X4 = a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) = X1 + 4X2 + 6X3 + 4X4 + 1 >= X1 + 0 = mark#(X1) mark#(U92(X1,X2,X3,X4)) = X1 + 4X2 + 6X3 + 4X4 + 1 >= X2 + 3X3 + 4X4 = a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) = 4X1 + 4X2 + 6X3 + 1X4 + 0 >= X1 + 0 = mark#(X1) mark#(U93(X1,X2,X3,X4)) = 4X1 + 4X2 + 6X3 + 1X4 + 0 >= X1 + X2 + X3 + X4 = a__U93#(mark(X1),X2,X3,X4) mark#(take(X1,X2)) = 6X1 + 4X2 + 2 >= X2 + 0 = mark#(X2) mark#(take(X1,X2)) = 6X1 + 4X2 + 2 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = 6X1 + 4X2 + 2 >= 6X1 + 4X2 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 2 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__zeros() = 2 >= 2 = cons(0(),zeros()) a__U11(tt()) = 5 >= 1 = tt() a__U21(tt()) = 1 >= 1 = tt() a__U31(tt()) = 1 >= 1 = tt() a__U41(tt(),V2) = V2 + 2 >= V2 + 0 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 1 >= 1 = tt() a__U51(tt(),V2) = V2 + 1 >= V2 = a__U52(a__isNatList(V2)) a__U52(tt()) = 1 >= 1 = tt() a__U61(tt(),V2) = 4V2 + 2 >= 2V2 + 2 = a__U62(a__isNatIList(V2)) a__U62(tt()) = 3 >= 1 = tt() a__U71(tt(),L,N) = 4L + N >= 4L + N = a__U72(a__isNat(N),L) a__U72(tt(),L) = 4L + 1 >= 4L = s(a__length(mark(L))) a__U81(tt()) = 3 >= 1 = nil() a__U91(tt(),IL,M,N) = 4IL + 6M + 4N + 1 >= 4IL + 6M + 4N + 1 = a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) = 4IL + 6M + 4N + 1 >= 4IL + 6M + 4N = a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) = 4IL + 6M + 1N + 5 >= 4IL + 6M + N + 2 = cons(mark(N),take(M,IL)) a__isNat(0()) = 1 >= 1 = tt() a__isNat(length(V1)) = 4V1 >= 4V1 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = V + 0 >= V = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 2 >= 1 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 + 2 >= V1 + V2 + 2 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 1 >= 1 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 + 2 >= V1 + V2 = a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) = 6V1 + 4V2 + 2 >= 1V1 + 4V2 = a__U61(a__isNat(V1),V2) a__length(nil()) = 5 >= 1 = 0() a__length(cons(N,L)) = 4L + 4N + 6 >= 4L + N = a__U71(a__isNatList(L),L,N) a__take(0(),IL) = 4IL + 7 >= 2IL + 2 = a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) = 4IL + 6M + 4N + 6 >= 4IL + 6M + 4N + 0 = a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) = 2 >= 2 = a__zeros() mark(U11(X)) = 4X >= 4X = 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 + 2 >= X1 + X2 + 2 = a__U41(mark(X1),X2) mark(U42(X)) = X + 0 >= X + 0 = 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 >= X = a__U52(mark(X)) mark(isNatList(X)) = X >= X = a__isNatList(X) mark(U61(X1,X2)) = 1X1 + 4X2 >= 1X1 + 4X2 = a__U61(mark(X1),X2) mark(U62(X)) = 2X >= 2X = a__U62(mark(X)) mark(U71(X1,X2,X3)) = 4X2 + X3 >= 4X2 + X3 = a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = a__U72(mark(X1),X2) mark(isNat(X)) = X >= X = a__isNat(X) mark(length(X)) = 4X >= 4X = a__length(mark(X)) mark(U81(X)) = 2X >= 2X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + 4X2 + 6X3 + 4X4 + 0 >= X1 + 4X2 + 6X3 + 4X4 + 0 = a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) = X1 + 4X2 + 6X3 + 4X4 + 1 >= X1 + 4X2 + 6X3 + 4X4 + 1 = a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) = 4X1 + 4X2 + 6X3 + 1X4 >= 4X1 + 4X2 + 6X3 + 1X4 = a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 6X1 + 4X2 + 2 >= 6X1 + 4X2 + 2 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(tt()) = 1 >= 1 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 1 >= 1 = nil() a__zeros() = 2 >= 2 = zeros() a__U11(X) = 4X >= 4X = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = X >= X = U31(X) a__U41(X1,X2) = X1 + X2 + 2 >= X1 + X2 + 2 = U41(X1,X2) a__U42(X) = X + 0 >= X + 0 = 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 >= X = U52(X) a__isNatList(X) = X >= X = isNatList(X) a__U61(X1,X2) = 1X1 + 4X2 >= 1X1 + 4X2 = U61(X1,X2) a__U62(X) = 2X >= 2X = U62(X) a__U71(X1,X2,X3) = 4X2 + X3 >= 4X2 + X3 = U71(X1,X2,X3) a__U72(X1,X2) = X1 + 4X2 >= X1 + 4X2 = U72(X1,X2) a__isNat(X) = X >= X = isNat(X) a__length(X) = 4X >= 4X = length(X) a__U81(X) = 2X >= 2X = U81(X) a__U91(X1,X2,X3,X4) = X1 + 4X2 + 6X3 + 4X4 + 0 >= X1 + 4X2 + 6X3 + 4X4 + 0 = U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) = X1 + 4X2 + 6X3 + 4X4 + 1 >= X1 + 4X2 + 6X3 + 4X4 + 1 = U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) = 4X1 + 4X2 + 6X3 + 1X4 >= 4X1 + 4X2 + 6X3 + 1X4 = U93(X1,X2,X3,X4) a__take(X1,X2) = 6X1 + 4X2 + 2 >= 6X1 + 4X2 + 2 = 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#(U72(X1,X2)) -> mark#(X1) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) 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#(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#(U72(X1,X2)) -> mark#(X1) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) 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#(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) = x0 + x1 + 0, [tt] = 4, [a__U62](x0) = 1x0 + 1, [a__U41](x0, x1) = x0 + x1 + 0, [0] = 4, [cons](x0, x1) = x0 + x1 + 4, [a__U71](x0, x1, x2) = x0 + x1 + x2 + 0, [s](x0) = x0, [U51](x0, x1) = x0 + x1, [a__U93#](x0, x1, x2, x3) = 1x0 + x2 + x3, [a__U93](x0, x1, x2, x3) = x0 + 1x1 + x2 + x3, [mark#](x0) = x0 + 0, [mark](x0) = x0 + 0, [U31](x0) = x0, [a__isNat](x0) = x0, [take](x0, x1) = x0 + 1x1 + 4, [U42](x0) = x0, [a__U81](x0) = x0 + 0, [length](x0) = x0 + 2, [U81](x0) = x0, [a__U11](x0) = x0 + 0, [U41](x0, x1) = x0 + x1 + 0, [a__U42](x0) = x0 + 0, [a__zeros] = 7, [U61](x0, x1) = x0 + 1x1 + 0, [a__U31](x0) = x0 + 0, [a__length](x0) = x0 + 2, [a__U92](x0, x1, x2, x3) = x0 + 1x1 + x2 + 1x3 + 0, [U11](x0) = x0, [a__U61](x0, x1) = x0 + 1x1 + 0, [U21](x0) = x0, [a__isNatList](x0) = x0 + 0, [a__take#](x0, x1) = x0 + 1x1 + 0, [U91](x0, x1, x2, x3) = x0 + 1x1 + x2 + 1x3 + 4, [nil] = 4, [a__U91](x0, x1, x2, x3) = x0 + 1x1 + x2 + 1x3 + 4, [a__U51](x0, x1) = x0 + x1, [a__isNatIList](x0) = x0 + 0, [a__U52](x0) = x0 + 0, [U62](x0) = 1x0 + 1, [a__U92#](x0, x1, x2, x3) = x0 + x2 + 1x3, [a__U21](x0) = x0, [isNatList](x0) = x0, [U93](x0, x1, x2, x3) = x0 + 1x1 + x2 + x3, [U52](x0) = x0 + 0, [U71](x0, x1, x2) = x0 + x1 + x2, [U72](x0, x1) = x0 + x1 + 0, [U92](x0, x1, x2, x3) = x0 + 1x1 + x2 + 1x3, [a__U91#](x0, x1, x2, x3) = x0 + x2 + 1x3 + 0, [isNatIList](x0) = x0, [a__take](x0, x1) = x0 + 1x1 + 4, [isNat](x0) = x0, [zeros] = 7 orientation: a__take#(s(M),cons(N,IL)) = 1IL + M + 1N + 5 >= IL + M + 1N + 0 = a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) = M + 1N + 4 >= M + 1N = a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) = M + 1N + 4 >= M + 1N = a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) = M + N + 5 >= N + 0 = mark#(N) mark#(U11(X)) = X + 0 >= X + 0 = mark#(X) mark#(U21(X)) = X + 0 >= X + 0 = mark#(X) mark#(U31(X)) = X + 0 >= X + 0 = mark#(X) mark#(U41(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(U42(X)) = X + 0 >= X + 0 = mark#(X) mark#(U51(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(U52(X)) = X + 0 >= X + 0 = mark#(X) mark#(U61(X1,X2)) = X1 + 1X2 + 0 >= X1 + 0 = mark#(X1) mark#(U62(X)) = 1X + 1 >= X + 0 = mark#(X) mark#(U72(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(U81(X)) = X + 0 >= X + 0 = mark#(X) mark#(U91(X1,X2,X3,X4)) = X1 + 1X2 + X3 + 1X4 + 4 >= X1 + 0 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = X1 + 1X2 + X3 + 1X4 + 4 >= X1 + X3 + 1X4 + 0 = a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) = X1 + 1X2 + X3 + 1X4 + 0 >= X1 + 0 = mark#(X1) mark#(U92(X1,X2,X3,X4)) = X1 + 1X2 + X3 + 1X4 + 0 >= X1 + X3 + 1X4 + 0 = a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) = X1 + 1X2 + X3 + X4 + 0 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = X1 + 1X2 + 4 >= X1 + 1X2 + 1 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 4 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__zeros() = 7 >= 7 = cons(0(),zeros()) a__U11(tt()) = 4 >= 4 = tt() a__U21(tt()) = 4 >= 4 = tt() a__U31(tt()) = 4 >= 4 = tt() a__U41(tt(),V2) = V2 + 4 >= V2 + 0 = a__U42(a__isNatIList(V2)) a__U42(tt()) = 4 >= 4 = tt() a__U51(tt(),V2) = V2 + 4 >= V2 + 0 = a__U52(a__isNatList(V2)) a__U52(tt()) = 4 >= 4 = tt() a__U61(tt(),V2) = 1V2 + 4 >= 1V2 + 1 = a__U62(a__isNatIList(V2)) a__U62(tt()) = 5 >= 4 = tt() a__U71(tt(),L,N) = L + N + 4 >= L + N + 0 = a__U72(a__isNat(N),L) a__U72(tt(),L) = L + 4 >= L + 2 = s(a__length(mark(L))) a__U81(tt()) = 4 >= 4 = nil() a__U91(tt(),IL,M,N) = 1IL + M + 1N + 4 >= 1IL + M + 1N + 0 = a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) = 1IL + M + 1N + 4 >= 1IL + M + N = a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) = 1IL + M + N + 4 >= 1IL + M + N + 4 = cons(mark(N),take(M,IL)) a__isNat(0()) = 4 >= 4 = tt() a__isNat(length(V1)) = V1 + 2 >= V1 + 0 = a__U11(a__isNatList(V1)) a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNat(V1)) a__isNatIList(V) = V + 0 >= V + 0 = a__U31(a__isNatList(V)) a__isNatIList(zeros()) = 7 >= 4 = tt() a__isNatIList(cons(V1,V2)) = V1 + V2 + 4 >= V1 + V2 + 0 = a__U41(a__isNat(V1),V2) a__isNatList(nil()) = 4 >= 4 = tt() a__isNatList(cons(V1,V2)) = V1 + V2 + 4 >= V1 + V2 = a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) = V1 + 1V2 + 4 >= V1 + 1V2 + 0 = a__U61(a__isNat(V1),V2) a__length(nil()) = 4 >= 4 = 0() a__length(cons(N,L)) = L + N + 4 >= L + N + 0 = a__U71(a__isNatList(L),L,N) a__take(0(),IL) = 1IL + 4 >= IL + 0 = a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) = 1IL + M + 1N + 5 >= 1IL + M + 1N + 4 = a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) = 7 >= 7 = a__zeros() mark(U11(X)) = X + 0 >= X + 0 = a__U11(mark(X)) mark(U21(X)) = X + 0 >= X + 0 = a__U21(mark(X)) mark(U31(X)) = X + 0 >= X + 0 = a__U31(mark(X)) mark(U41(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U41(mark(X1),X2) mark(U42(X)) = X + 0 >= X + 0 = a__U42(mark(X)) mark(isNatIList(X)) = X + 0 >= X + 0 = a__isNatIList(X) mark(U51(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = 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 + 1X2 + 0 >= X1 + 1X2 + 0 = a__U61(mark(X1),X2) mark(U62(X)) = 1X + 1 >= 1X + 1 = a__U62(mark(X)) mark(U71(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__U72(mark(X1),X2) mark(isNat(X)) = X + 0 >= X = a__isNat(X) mark(length(X)) = X + 2 >= X + 2 = a__length(mark(X)) mark(U81(X)) = X + 0 >= X + 0 = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = X1 + 1X2 + X3 + 1X4 + 4 >= X1 + 1X2 + X3 + 1X4 + 4 = a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) = X1 + 1X2 + X3 + 1X4 + 0 >= X1 + 1X2 + X3 + 1X4 + 0 = a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) = X1 + 1X2 + X3 + X4 + 0 >= X1 + 1X2 + X3 + X4 + 0 = a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) = X1 + 1X2 + 4 >= X1 + 1X2 + 4 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = cons(mark(X1),X2) mark(0()) = 4 >= 4 = 0() mark(tt()) = 4 >= 4 = tt() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 4 >= 4 = nil() a__zeros() = 7 >= 7 = zeros() a__U11(X) = X + 0 >= X = U11(X) a__U21(X) = X >= X = U21(X) a__U31(X) = X + 0 >= X = U31(X) a__U41(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U41(X1,X2) a__U42(X) = X + 0 >= X = U42(X) a__isNatIList(X) = X + 0 >= X = 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 = isNatList(X) a__U61(X1,X2) = X1 + 1X2 + 0 >= X1 + 1X2 + 0 = U61(X1,X2) a__U62(X) = 1X + 1 >= 1X + 1 = U62(X) a__U71(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 = U71(X1,X2,X3) a__U72(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = U72(X1,X2) a__isNat(X) = X >= X = isNat(X) a__length(X) = X + 2 >= X + 2 = length(X) a__U81(X) = X + 0 >= X = U81(X) a__U91(X1,X2,X3,X4) = X1 + 1X2 + X3 + 1X4 + 4 >= X1 + 1X2 + X3 + 1X4 + 4 = U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) = X1 + 1X2 + X3 + 1X4 + 0 >= X1 + 1X2 + X3 + 1X4 = U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) = X1 + 1X2 + X3 + X4 >= X1 + 1X2 + X3 + X4 = U93(X1,X2,X3,X4) a__take(X1,X2) = X1 + 1X2 + 4 >= X1 + 1X2 + 4 = 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#(U72(X1,X2)) -> mark#(X1) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) 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#(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#(U72(X1,X2)) -> mark#(X1) mark#(U81(X)) -> mark#(X) mark#(U91(X1,X2,X3,X4)) -> mark#(X1) mark#(U91(X1,X2,X3,X4)) -> a__U91#(mark(X1),X2,X3,X4) 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#(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) = x0 + 2, [tt] = 0, [a__U62](x0) = 0, [a__U41](x0, x1) = x0, [0] = 0, [cons](x0, x1) = x0, [a__U71](x0, x1, x2) = 2, [s](x0) = x0, [U51](x0, x1) = x0, [a__U93#](x0, x1, x2, x3) = 1/2x3, [a__U93](x0, x1, x2, x3) = x0 + 2x3, [mark#](x0) = 1/2x0, [mark](x0) = 2x0, [U31](x0) = x0, [a__isNat](x0) = 0, [take](x0, x1) = 2x1, [U42](x0) = x0, [a__U81](x0) = x0, [length](x0) = 2, [U81](x0) = x0, [a__U11](x0) = 2x0, [U41](x0, x1) = x0, [a__U42](x0) = x0, [a__zeros] = 0, [U61](x0, x1) = 2x0, [a__U31](x0) = x0, [a__length](x0) = 2, [a__U92](x0, x1, x2, x3) = 2x0 + 2x3, [U11](x0) = 2x0, [a__U61](x0, x1) = 2x0, [U21](x0) = 2x0, [a__isNatList](x0) = 0, [a__take#](x0, x1) = 1/2x1, [U91](x0, x1, x2, x3) = x0 + 2x3, [nil] = 0, [a__U91](x0, x1, x2, x3) = x0 + 2x3, [a__U51](x0, x1) = x0, [a__isNatIList](x0) = 0, [a__U52](x0) = x0, [U62](x0) = 0, [a__U92#](x0, x1, x2, x3) = 1/2x3, [a__U21](x0) = 2x0, [isNatList](x0) = 0, [U93](x0, x1, x2, x3) = x0 + x3, [U52](x0) = x0, [U71](x0, x1, x2) = 3/2, [U72](x0, x1) = x0 + 2, [U92](x0, x1, x2, x3) = 2x0 + 2x3, [a__U91#](x0, x1, x2, x3) = 1/2x3, [isNatIList](x0) = 0, [a__take](x0, x1) = 2x1, [isNat](x0) = 0, [zeros] = 0 orientation: a__take#(s(M),cons(N,IL)) = 1/2N >= 1/2N = a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) = 1/2N >= 1/2N = a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) = 1/2N >= 1/2N = a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) = 1/2N >= 1/2N = mark#(N) mark#(U11(X)) = X >= 1/2X = mark#(X) mark#(U21(X)) = X >= 1/2X = mark#(X) mark#(U31(X)) = 1/2X >= 1/2X = mark#(X) mark#(U41(X1,X2)) = 1/2X1 >= 1/2X1 = mark#(X1) mark#(U42(X)) = 1/2X >= 1/2X = mark#(X) mark#(U51(X1,X2)) = 1/2X1 >= 1/2X1 = mark#(X1) mark#(U52(X)) = 1/2X >= 1/2X = mark#(X) mark#(U61(X1,X2)) = X1 >= 1/2X1 = mark#(X1) mark#(U72(X1,X2)) = 1/2X1 + 1 >= 1/2X1 = mark#(X1) mark#(U81(X)) = 1/2X >= 1/2X = mark#(X) mark#(U91(X1,X2,X3,X4)) = 1/2X1 + X4 >= 1/2X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = 1/2X1 + X4 >= 1/2X4 = a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) = X1 + X4 >= 1/2X1 = mark#(X1) mark#(U92(X1,X2,X3,X4)) = X1 + X4 >= 1/2X4 = a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) = 1/2X1 + 1/2X4 >= 1/2X1 = mark#(X1) mark#(take(X1,X2)) = X2 >= X2 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = 1/2X1 >= 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) = 2 >= 2 = a__U72(a__isNat(N),L) a__U72(tt(),L) = 2 >= 2 = s(a__length(mark(L))) a__U81(tt()) = 0 >= 0 = nil() a__U91(tt(),IL,M,N) = 2N >= 2N = a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) = 2N >= 2N = a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) = 2N >= 2N = 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()) = 2 >= 0 = 0() a__length(cons(N,L)) = 2 >= 2 = 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)) = 2N >= 2N = a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) = 0 >= 0 = a__zeros() mark(U11(X)) = 4X >= 4X = a__U11(mark(X)) mark(U21(X)) = 4X >= 4X = a__U21(mark(X)) mark(U31(X)) = 2X >= 2X = a__U31(mark(X)) mark(U41(X1,X2)) = 2X1 >= 2X1 = 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)) = 2X1 >= 2X1 = 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)) = 4X1 >= 4X1 = a__U61(mark(X1),X2) mark(U62(X)) = 0 >= 0 = a__U62(mark(X)) mark(U71(X1,X2,X3)) = 3 >= 2 = a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) = 2X1 + 4 >= 2X1 + 2 = a__U72(mark(X1),X2) mark(isNat(X)) = 0 >= 0 = a__isNat(X) mark(length(X)) = 4 >= 2 = a__length(mark(X)) mark(U81(X)) = 2X >= 2X = a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) = 2X1 + 4X4 >= 2X1 + 2X4 = a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) = 4X1 + 4X4 >= 4X1 + 2X4 = a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) = 2X1 + 2X4 >= 2X1 + 2X4 = a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) = 4X2 >= 4X2 = a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) = 2X1 >= 2X1 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 2X >= 2X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() a__zeros() = 0 >= 0 = zeros() a__U11(X) = 2X >= 2X = U11(X) a__U21(X) = 2X >= 2X = U21(X) a__U31(X) = X >= X = U31(X) a__U41(X1,X2) = X1 >= X1 = U41(X1,X2) a__U42(X) = X >= X = U42(X) a__isNatIList(X) = 0 >= 0 = isNatIList(X) a__U51(X1,X2) = X1 >= X1 = U51(X1,X2) a__U52(X) = X >= X = U52(X) a__isNatList(X) = 0 >= 0 = isNatList(X) a__U61(X1,X2) = 2X1 >= 2X1 = U61(X1,X2) a__U62(X) = 0 >= 0 = U62(X) a__U71(X1,X2,X3) = 2 >= 3/2 = U71(X1,X2,X3) a__U72(X1,X2) = X1 + 2 >= X1 + 2 = U72(X1,X2) a__isNat(X) = 0 >= 0 = isNat(X) a__length(X) = 2 >= 2 = length(X) a__U81(X) = X >= X = U81(X) a__U91(X1,X2,X3,X4) = X1 + 2X4 >= X1 + 2X4 = U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) = 2X1 + 2X4 >= 2X1 + 2X4 = U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) = X1 + 2X4 >= X1 + X4 = U93(X1,X2,X3,X4) a__take(X1,X2) = 2X2 >= 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#(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#(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#(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#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dim=3 usable rules: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) interpretation: [0] [a__U72](x0, x1) = [0] [0], [0] [tt] = [0] [0], [0] [a__U62](x0) = [0] [0], [1 0 1] [a__U41](x0, x1) = [0 1 0]x0 [0 0 0] , [0] [0] = [0] [0], [0 0 0] [0 0 0] [cons](x0, x1) = [1 1 0]x0 + [1 0 1]x1 [0 0 0] [0 0 0] , [0 0 0] [a__U71](x0, x1, x2) = [1 0 0]x0 [0 0 0] , [1 0 0] [s](x0) = [0 1 0]x0 [1 0 0] , [1 0 0] [U51](x0, x1) = [1 1 0]x0 [0 0 0] , [a__U93#](x0, x1, x2, x3) = [1 1 0]x3, [1 0 0] [0 0 0] [0 0 0] [a__U93](x0, x1, x2, x3) = [1 1 0]x0 + [1 0 1]x1 + [0 1 0]x3 [0 0 0] [0 0 0] [0 0 0] , [mark#](x0) = [1 1 0]x0, [0 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [U31](x0) = [1 1 0]x0 [0 0 0] , [0] [a__isNat](x0) = [0] [0], [1 0 0] [0] [take](x0, x1) = [1 1 0]x1 + [1] [0 0 0] [0], [0 0 0] [U42](x0) = [1 1 0]x0 [0 0 0] , [1 0 0] [0] [a__U81](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [0] [length](x0) = [0] [0], [1 0 0] [0] [U81](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [a__U11](x0) = [0 1 0]x0 [0 0 0] , [1 0 1] [U41](x0, x1) = [0 1 0]x0 [0 0 0] , [0 0 0] [a__U42](x0) = [1 1 1]x0 [0 0 0] , [0] [a__zeros] = [0] [0], [1 0 0] [U61](x0, x1) = [0 1 0]x0 [0 0 0] , [1 0 0] [a__U31](x0) = [1 1 0]x0 [0 0 0] , [0] [a__length](x0) = [0] [0], [1 0 1] [0 0 0] [0 0 0] [a__U92](x0, x1, x2, x3) = [1 1 1]x0 + [1 0 1]x1 + [1 1 0]x3 [0 0 0] [0 0 0] [0 0 0] , [1 0 0] [U11](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [a__U61](x0, x1) = [0 1 0]x0 [0 0 0] , [1 0 1] [U21](x0) = [0 1 0]x0 [0 0 0] , [0] [a__isNatList](x0) = [0] [0], [a__take#](x0, x1) = [0 0 1]x0 + [0 1 0]x1, [1 0 1] [0 0 0] [0 0 0] [U91](x0, x1, x2, x3) = [1 1 0]x0 + [1 0 1]x1 + [1 1 0]x3 [0 0 0] [0 0 0] [0 0 0] , [0] [nil] = [0] [0], [1 0 1] [0 0 0] [0 0 0] [a__U91](x0, x1, x2, x3) = [1 1 0]x0 + [1 0 1]x1 + [1 1 0]x3 [0 0 0] [0 0 0] [0 0 0] , [1 0 0] [a__U51](x0, x1) = [1 1 0]x0 [0 0 0] , [0] [a__isNatIList](x0) = [0] [0], [1 0 0] [a__U52](x0) = [0 1 0]x0 [0 0 0] , [0] [U62](x0) = [0] [0], [a__U92#](x0, x1, x2, x3) = [1 0 0]x1 + [1 1 0]x3, [1 0 1] [a__U21](x0) = [0 1 0]x0 [0 0 0] , [0] [isNatList](x0) = [0] [0], [0 0 0] [0 0 0] [0 0 0] [U93](x0, x1, x2, x3) = [1 1 0]x0 + [1 0 1]x1 + [0 1 0]x3 [0 0 0] [0 0 0] [0 0 0] , [1 0 0] [U52](x0) = [0 1 0]x0 [0 0 0] , [0 0 0] [U71](x0, x1, x2) = [1 0 0]x0 [0 0 0] , [0] [U72](x0, x1) = [0] [0], [1 0 1] [0 0 0] [0 0 0] [U92](x0, x1, x2, x3) = [1 1 1]x0 + [1 0 1]x1 + [1 1 0]x3 [0 0 0] [0 0 0] [0 0 0] , [a__U91#](x0, x1, x2, x3) = [1 0 0]x1 + [1 1 0]x3, [0] [isNatIList](x0) = [0] [0], [1 0 0] [0] [a__take](x0, x1) = [1 1 1]x1 + [1] [0 0 0] [0], [0] [isNat](x0) = [0] [0], [0] [zeros] = [0] [0] orientation: a__take#(s(M),cons(N,IL)) = [1 0 1]IL + [1 0 0]M + [1 1 0]N >= [1 0 0]IL + [1 1 0]N = a__U91#(a__isNatIList(IL),IL,M,N) a__U91#(tt(),IL,M,N) = [1 0 0]IL + [1 1 0]N >= [1 0 0]IL + [1 1 0]N = a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) = [1 0 0]IL + [1 1 0]N >= [1 1 0]N = a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) = [1 1 0]N >= [1 1 0]N = mark#(N) mark#(U11(X)) = [1 1 0]X >= [1 1 0]X = mark#(X) mark#(U21(X)) = [1 1 1]X >= [1 1 0]X = mark#(X) mark#(U31(X)) = [2 1 0]X >= [1 1 0]X = mark#(X) mark#(U41(X1,X2)) = [1 1 1]X1 >= [1 1 0]X1 = mark#(X1) mark#(U42(X)) = [1 1 0]X >= [1 1 0]X = mark#(X) mark#(U51(X1,X2)) = [2 1 0]X1 >= [1 1 0]X1 = mark#(X1) mark#(U52(X)) = [1 1 0]X >= [1 1 0]X = mark#(X) mark#(U61(X1,X2)) = [1 1 0]X1 >= [1 1 0]X1 = mark#(X1) mark#(U81(X)) = [1 1 0]X + [1] >= [1 1 0]X = mark#(X) mark#(U91(X1,X2,X3,X4)) = [2 1 1]X1 + [1 0 1]X2 + [1 1 0]X4 >= [1 1 0]X1 = mark#(X1) mark#(U91(X1,X2,X3,X4)) = [2 1 1]X1 + [1 0 1]X2 + [1 1 0]X4 >= [1 0 0]X2 + [1 1 0]X4 = a__U91#(mark(X1),X2,X3,X4) mark#(U92(X1,X2,X3,X4)) = [2 1 2]X1 + [1 0 1]X2 + [1 1 0]X4 >= [1 1 0]X1 = mark#(X1) mark#(U92(X1,X2,X3,X4)) = [2 1 2]X1 + [1 0 1]X2 + [1 1 0]X4 >= [1 0 0]X2 + [1 1 0]X4 = a__U92#(mark(X1),X2,X3,X4) mark#(U93(X1,X2,X3,X4)) = [1 1 0]X1 + [1 0 1]X2 + [0 1 0]X4 >= [1 1 0]X1 = mark#(X1) mark#(take(X1,X2)) = [2 1 0]X2 + [1] >= [0 1 0]X2 = a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = [1 1 0]X1 + [1 0 1]X2 >= [1 1 0]X1 = mark#(X1) mark#(s(X)) = [1 1 0]X >= [1 1 0]X = mark#(X) [0] [0] a__zeros() = [0] >= [0] = cons(0(),zeros()) [0] [0] [0] [0] a__U11(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U21(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U31(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U41(tt(),V2) = [0] >= [0] = a__U42(a__isNatIList(V2)) [0] [0] [0] [0] a__U42(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U51(tt(),V2) = [0] >= [0] = a__U52(a__isNatList(V2)) [0] [0] [0] [0] a__U52(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U61(tt(),V2) = [0] >= [0] = a__U62(a__isNatIList(V2)) [0] [0] [0] [0] a__U62(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] a__U71(tt(),L,N) = [0] >= [0] = a__U72(a__isNat(N),L) [0] [0] [0] [0] a__U72(tt(),L) = [0] >= [0] = s(a__length(mark(L))) [0] [0] [0] [0] a__U81(tt()) = [1] >= [0] = nil() [0] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U91(tt(),IL,M,N) = [1 0 1]IL + [1 1 0]N >= [1 0 1]IL + [1 1 0]N = a__U92(a__isNat(M),IL,M,N) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U92(tt(),IL,M,N) = [1 0 1]IL + [1 1 0]N >= [1 0 1]IL + [0 1 0]N = a__U93(a__isNat(N),IL,M,N) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U93(tt(),IL,M,N) = [1 0 1]IL + [0 1 0]N >= [1 0 0]IL + [0 1 0]N = cons(mark(N),take(M,IL)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] a__isNat(0()) = [0] >= [0] = tt() [0] [0] [0] [0] a__isNat(length(V1)) = [0] >= [0] = a__U11(a__isNatList(V1)) [0] [0] [0] [0] a__isNat(s(V1)) = [0] >= [0] = a__U21(a__isNat(V1)) [0] [0] [0] [0] a__isNatIList(V) = [0] >= [0] = a__U31(a__isNatList(V)) [0] [0] [0] [0] a__isNatIList(zeros()) = [0] >= [0] = tt() [0] [0] [0] [0] a__isNatIList(cons(V1,V2)) = [0] >= [0] = a__U41(a__isNat(V1),V2) [0] [0] [0] [0] a__isNatList(nil()) = [0] >= [0] = tt() [0] [0] [0] [0] a__isNatList(cons(V1,V2)) = [0] >= [0] = a__U51(a__isNat(V1),V2) [0] [0] [0] [0] a__isNatList(take(V1,V2)) = [0] >= [0] = a__U61(a__isNat(V1),V2) [0] [0] [0] [0] a__length(nil()) = [0] >= [0] = 0() [0] [0] [0] [0] a__length(cons(N,L)) = [0] >= [0] = a__U71(a__isNatList(L),L,N) [0] [0] [1 0 0] [0] [0] a__take(0(),IL) = [1 1 1]IL + [1] >= [1] = a__U81(a__isNatIList(IL)) [0 0 0] [0] [0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] a__take(s(M),cons(N,IL)) = [1 0 1]IL + [1 1 0]N + [1] >= [1 0 1]IL + [1 1 0]N = a__U91(a__isNatIList(IL),IL,M,N) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] mark(zeros()) = [0] >= [0] = a__zeros() [0] [0] [0 0 0] [0 0 0] mark(U11(X)) = [0 1 0]X >= [0 1 0]X = a__U11(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U21(X)) = [0 1 0]X >= [0 1 0]X = a__U21(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U31(X)) = [1 1 0]X >= [0 1 0]X = a__U31(mark(X)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U41(X1,X2)) = [0 1 0]X1 >= [0 1 0]X1 = a__U41(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U42(X)) = [1 1 0]X >= [0 1 0]X = a__U42(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(isNatIList(X)) = [0] >= [0] = a__isNatIList(X) [0] [0] [0 0 0] [0 0 0] mark(U51(X1,X2)) = [1 1 0]X1 >= [0 1 0]X1 = a__U51(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U52(X)) = [0 1 0]X >= [0 1 0]X = a__U52(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(isNatList(X)) = [0] >= [0] = a__isNatList(X) [0] [0] [0 0 0] [0 0 0] mark(U61(X1,X2)) = [0 1 0]X1 >= [0 1 0]X1 = a__U61(mark(X1),X2) [0 0 0] [0 0 0] [0] [0] mark(U62(X)) = [0] >= [0] = a__U62(mark(X)) [0] [0] [0 0 0] [0] mark(U71(X1,X2,X3)) = [1 0 0]X1 >= [0] = a__U71(mark(X1),X2,X3) [0 0 0] [0] [0] [0] mark(U72(X1,X2)) = [0] >= [0] = a__U72(mark(X1),X2) [0] [0] [0] [0] mark(isNat(X)) = [0] >= [0] = a__isNat(X) [0] [0] [0] [0] mark(length(X)) = [0] >= [0] = a__length(mark(X)) [0] [0] [0 0 0] [0] [0 0 0] [0] mark(U81(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__U81(mark(X)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U91(X1,X2,X3,X4)) = [1 1 0]X1 + [1 0 1]X2 + [1 1 0]X4 >= [0 1 0]X1 + [1 0 1]X2 + [1 1 0]X4 = a__U91(mark(X1),X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U92(X1,X2,X3,X4)) = [1 1 1]X1 + [1 0 1]X2 + [1 1 0]X4 >= [0 1 0]X1 + [1 0 1]X2 + [1 1 0]X4 = a__U92(mark(X1),X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(U93(X1,X2,X3,X4)) = [1 1 0]X1 + [1 0 1]X2 + [0 1 0]X4 >= [0 1 0]X1 + [1 0 1]X2 + [0 1 0]X4 = a__U93(mark(X1),X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0] mark(take(X1,X2)) = [1 1 0]X2 + [1] >= [0 1 0]X2 + [1] = a__take(mark(X1),mark(X2)) [0 0 0] [0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] mark(cons(X1,X2)) = [1 1 0]X1 + [1 0 1]X2 >= [0 1 0]X1 + [1 0 1]X2 = cons(mark(X1),X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [0 0 0] [0 0 0] mark(s(X)) = [0 1 0]X >= [0 1 0]X = s(mark(X)) [0 0 0] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [0] [0] a__zeros() = [0] >= [0] = zeros() [0] [0] [1 0 0] [1 0 0] a__U11(X) = [0 1 0]X >= [0 1 0]X = U11(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] a__U21(X) = [0 1 0]X >= [0 1 0]X = U21(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U31(X) = [1 1 0]X >= [1 1 0]X = U31(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] a__U41(X1,X2) = [0 1 0]X1 >= [0 1 0]X1 = U41(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U42(X) = [1 1 1]X >= [1 1 0]X = U42(X) [0 0 0] [0 0 0] [0] [0] a__isNatIList(X) = [0] >= [0] = isNatIList(X) [0] [0] [1 0 0] [1 0 0] a__U51(X1,X2) = [1 1 0]X1 >= [1 1 0]X1 = U51(X1,X2) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U52(X) = [0 1 0]X >= [0 1 0]X = U52(X) [0 0 0] [0 0 0] [0] [0] a__isNatList(X) = [0] >= [0] = isNatList(X) [0] [0] [1 0 0] [1 0 0] a__U61(X1,X2) = [0 1 0]X1 >= [0 1 0]X1 = U61(X1,X2) [0 0 0] [0 0 0] [0] [0] a__U62(X) = [0] >= [0] = U62(X) [0] [0] [0 0 0] [0 0 0] a__U71(X1,X2,X3) = [1 0 0]X1 >= [1 0 0]X1 = U71(X1,X2,X3) [0 0 0] [0 0 0] [0] [0] a__U72(X1,X2) = [0] >= [0] = U72(X1,X2) [0] [0] [0] [0] a__isNat(X) = [0] >= [0] = isNat(X) [0] [0] [0] [0] a__length(X) = [0] >= [0] = length(X) [0] [0] [1 0 0] [0] [1 0 0] [0] a__U81(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = U81(X) [0 0 0] [0] [0 0 0] [0] [1 0 1] [0 0 0] [0 0 0] [1 0 1] [0 0 0] [0 0 0] a__U91(X1,X2,X3,X4) = [1 1 0]X1 + [1 0 1]X2 + [1 1 0]X4 >= [1 1 0]X1 + [1 0 1]X2 + [1 1 0]X4 = U91(X1,X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [0 0 0] [0 0 0] [1 0 1] [0 0 0] [0 0 0] a__U92(X1,X2,X3,X4) = [1 1 1]X1 + [1 0 1]X2 + [1 1 0]X4 >= [1 1 1]X1 + [1 0 1]X2 + [1 1 0]X4 = U92(X1,X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] a__U93(X1,X2,X3,X4) = [1 1 0]X1 + [1 0 1]X2 + [0 1 0]X4 >= [1 1 0]X1 + [1 0 1]X2 + [0 1 0]X4 = U93(X1,X2,X3,X4) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] a__take(X1,X2) = [1 1 1]X2 + [1] >= [1 1 0]X2 + [1] = take(X1,X2) [0 0 0] [0] [0 0 0] [0] 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#(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#(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#(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#(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: 1 #rules: 18 #arcs: 425/361 DPs: a__U91#(tt(),IL,M,N) -> a__U92#(a__isNat(M),IL,M,N) a__U92#(tt(),IL,M,N) -> a__U93#(a__isNat(N),IL,M,N) a__U93#(tt(),IL,M,N) -> mark#(N) mark#(U11(X)) -> mark#(X) mark#(U21(X)) -> mark#(X) mark#(U31(X)) -> mark#(X) mark#(U41(X1,X2)) -> mark#(X1) mark#(U42(X)) -> mark#(X) mark#(U51(X1,X2)) -> mark#(X1) mark#(U52(X)) -> mark#(X) mark#(U61(X1,X2)) -> mark#(X1) mark#(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#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Subterm Criterion Processor: simple projection: pi(mark#) = 0 pi(a__U91#) = 3 pi(a__U92#) = 3 pi(a__U93#) = 3 problem: DPs: a__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) 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: 214/9 DPs: a__U71#(tt(),L,N) -> a__U72#(a__isNat(N),L) a__U72#(tt(),L) -> a__length#(mark(L)) a__length#(cons(N,L)) -> a__U71#(a__isNatList(L),L,N) TRS: a__zeros() -> cons(0(),zeros()) a__U11(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt()) -> tt() a__U41(tt(),V2) -> a__U42(a__isNatIList(V2)) a__U42(tt()) -> tt() a__U51(tt(),V2) -> a__U52(a__isNatList(V2)) a__U52(tt()) -> tt() a__U61(tt(),V2) -> a__U62(a__isNatIList(V2)) a__U62(tt()) -> tt() a__U71(tt(),L,N) -> a__U72(a__isNat(N),L) a__U72(tt(),L) -> s(a__length(mark(L))) a__U81(tt()) -> nil() a__U91(tt(),IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U92(tt(),IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U93(tt(),IL,M,N) -> cons(mark(N),take(M,IL)) a__isNat(0()) -> tt() a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(zeros()) -> tt() a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatList(nil()) -> tt() a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__length(nil()) -> 0() a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__take(0(),IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) mark(zeros()) -> a__zeros() mark(U11(X)) -> a__U11(mark(X)) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U42(X)) -> a__U42(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U61(X1,X2)) -> a__U61(mark(X1),X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2)) -> a__U72(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1,X2,X3,X4)) -> a__U91(mark(X1),X2,X3,X4) mark(U92(X1,X2,X3,X4)) -> a__U92(mark(X1),X2,X3,X4) mark(U93(X1,X2,X3,X4)) -> a__U93(mark(X1),X2,X3,X4) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() a__zeros() -> zeros() a__U11(X) -> U11(X) a__U21(X) -> U21(X) a__U31(X) -> U31(X) a__U41(X1,X2) -> U41(X1,X2) a__U42(X) -> U42(X) a__isNatIList(X) -> isNatIList(X) a__U51(X1,X2) -> U51(X1,X2) a__U52(X) -> U52(X) a__isNatList(X) -> isNatList(X) a__U61(X1,X2) -> U61(X1,X2) a__U62(X) -> U62(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2) -> U72(X1,X2) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__U81(X) -> U81(X) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__take(X1,X2) -> take(X1,X2) Matrix Interpretation Processor: dim=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