/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR IL L M N V V1 V2 X X1 X2 X3 X4) (RULES a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ) Problem 1: Dependency Pairs Processor: -> Pairs: A__U41(tt,V2) -> A__U42(a__isNatIList(V2)) A__U41(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V2) -> A__U52(a__isNatList(V2)) A__U51(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,V2) -> A__U62(a__isNatIList(V2)) A__U61(tt,V2) -> A__ISNATILIST(V2) A__U71(tt,L,N) -> A__U72(a__isNat(N),L) A__U71(tt,L,N) -> A__ISNAT(N) A__U72(tt,L) -> A__LENGTH(mark(L)) A__U72(tt,L) -> MARK(L) A__U91(tt,IL,M,N) -> A__U92(a__isNat(M),IL,M,N) A__U91(tt,IL,M,N) -> A__ISNAT(M) A__U92(tt,IL,M,N) -> A__U93(a__isNat(N),IL,M,N) A__U92(tt,IL,M,N) -> A__ISNAT(N) A__U93(tt,IL,M,N) -> MARK(N) A__ISNAT(length(V1)) -> A__U11(a__isNatList(V1)) A__ISNAT(length(V1)) -> A__ISNATLIST(V1) A__ISNAT(s(V1)) -> A__U21(a__isNat(V1)) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNat(V1),V2) A__ISNATILIST(cons(V1,V2)) -> A__ISNAT(V1) A__ISNATILIST(V) -> A__U31(a__isNatList(V)) A__ISNATILIST(V) -> A__ISNATLIST(V) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__isNat(V1),V2) A__ISNATLIST(cons(V1,V2)) -> A__ISNAT(V1) A__ISNATLIST(take(V1,V2)) -> A__U61(a__isNat(V1),V2) A__ISNATLIST(take(V1,V2)) -> A__ISNAT(V1) A__LENGTH(cons(N,L)) -> A__U71(a__isNatList(L),L,N) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) A__TAKE(0,IL) -> A__U81(a__isNatIList(IL)) A__TAKE(0,IL) -> A__ISNATILIST(IL) A__TAKE(s(M),cons(N,IL)) -> A__U91(a__isNatIList(IL),IL,M,N) A__TAKE(s(M),cons(N,IL)) -> A__ISNATILIST(IL) MARK(U11(X)) -> A__U11(mark(X)) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> A__U21(mark(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> A__U31(mark(X)) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> A__U41(mark(X1),X2) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> A__U42(mark(X)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> A__U51(mark(X1),X2) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> A__U52(mark(X)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(U62(X)) -> A__U62(mark(X)) MARK(U62(X)) -> MARK(X) MARK(U71(X1,X2,X3)) -> A__U71(mark(X1),X2,X3) MARK(U71(X1,X2,X3)) -> MARK(X1) MARK(U72(X1,X2)) -> A__U72(mark(X1),X2) MARK(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> A__U81(mark(X)) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> A__U91(mark(X1),X2,X3,X4) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> A__TAKE(mark(X1),mark(X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) MARK(zeros) -> A__ZEROS -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1: SCC Processor: -> Pairs: A__U41(tt,V2) -> A__U42(a__isNatIList(V2)) A__U41(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V2) -> A__U52(a__isNatList(V2)) A__U51(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,V2) -> A__U62(a__isNatIList(V2)) A__U61(tt,V2) -> A__ISNATILIST(V2) A__U71(tt,L,N) -> A__U72(a__isNat(N),L) A__U71(tt,L,N) -> A__ISNAT(N) A__U72(tt,L) -> A__LENGTH(mark(L)) A__U72(tt,L) -> MARK(L) A__U91(tt,IL,M,N) -> A__U92(a__isNat(M),IL,M,N) A__U91(tt,IL,M,N) -> A__ISNAT(M) A__U92(tt,IL,M,N) -> A__U93(a__isNat(N),IL,M,N) A__U92(tt,IL,M,N) -> A__ISNAT(N) A__U93(tt,IL,M,N) -> MARK(N) A__ISNAT(length(V1)) -> A__U11(a__isNatList(V1)) A__ISNAT(length(V1)) -> A__ISNATLIST(V1) A__ISNAT(s(V1)) -> A__U21(a__isNat(V1)) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNat(V1),V2) A__ISNATILIST(cons(V1,V2)) -> A__ISNAT(V1) A__ISNATILIST(V) -> A__U31(a__isNatList(V)) A__ISNATILIST(V) -> A__ISNATLIST(V) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__isNat(V1),V2) A__ISNATLIST(cons(V1,V2)) -> A__ISNAT(V1) A__ISNATLIST(take(V1,V2)) -> A__U61(a__isNat(V1),V2) A__ISNATLIST(take(V1,V2)) -> A__ISNAT(V1) A__LENGTH(cons(N,L)) -> A__U71(a__isNatList(L),L,N) A__LENGTH(cons(N,L)) -> A__ISNATLIST(L) A__TAKE(0,IL) -> A__U81(a__isNatIList(IL)) A__TAKE(0,IL) -> A__ISNATILIST(IL) A__TAKE(s(M),cons(N,IL)) -> A__U91(a__isNatIList(IL),IL,M,N) A__TAKE(s(M),cons(N,IL)) -> A__ISNATILIST(IL) MARK(U11(X)) -> A__U11(mark(X)) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> A__U21(mark(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> A__U31(mark(X)) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> A__U41(mark(X1),X2) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> A__U42(mark(X)) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> A__U51(mark(X1),X2) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> A__U52(mark(X)) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2)) -> A__U61(mark(X1),X2) MARK(U61(X1,X2)) -> MARK(X1) MARK(U62(X)) -> A__U62(mark(X)) MARK(U62(X)) -> MARK(X) MARK(U71(X1,X2,X3)) -> A__U71(mark(X1),X2,X3) MARK(U71(X1,X2,X3)) -> MARK(X1) MARK(U72(X1,X2)) -> A__U72(mark(X1),X2) MARK(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> A__U81(mark(X)) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> A__U91(mark(X1),X2,X3,X4) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> A__TAKE(mark(X1),mark(X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) MARK(zeros) -> A__ZEROS -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U41(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,V2) -> A__ISNATILIST(V2) A__ISNAT(length(V1)) -> A__ISNATLIST(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNat(V1),V2) A__ISNATILIST(cons(V1,V2)) -> A__ISNAT(V1) A__ISNATILIST(V) -> A__ISNATLIST(V) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__isNat(V1),V2) A__ISNATLIST(cons(V1,V2)) -> A__ISNAT(V1) A__ISNATLIST(take(V1,V2)) -> A__U61(a__isNat(V1),V2) A__ISNATLIST(take(V1,V2)) -> A__ISNAT(V1) ->->-> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: A__U71(tt,L,N) -> A__U72(a__isNat(N),L) A__U72(tt,L) -> A__LENGTH(mark(L)) A__U72(tt,L) -> MARK(L) 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) A__LENGTH(cons(N,L)) -> A__U71(a__isNatList(L),L,N) A__TAKE(s(M),cons(N,IL)) -> A__U91(a__isNatIList(IL),IL,M,N) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1,X2,X3)) -> A__U71(mark(X1),X2,X3) MARK(U71(X1,X2,X3)) -> MARK(X1) MARK(U72(X1,X2)) -> A__U72(mark(X1),X2) MARK(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> A__U91(mark(X1),X2,X3,X4) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> A__TAKE(mark(X1),mark(X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: A__U41(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,V2) -> A__ISNATILIST(V2) A__ISNAT(length(V1)) -> A__ISNATLIST(V1) A__ISNAT(s(V1)) -> A__ISNAT(V1) A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNat(V1),V2) A__ISNATILIST(cons(V1,V2)) -> A__ISNAT(V1) A__ISNATILIST(V) -> A__ISNATLIST(V) A__ISNATLIST(cons(V1,V2)) -> A__U51(a__isNat(V1),V2) A__ISNATLIST(cons(V1,V2)) -> A__ISNAT(V1) A__ISNATLIST(take(V1,V2)) -> A__U61(a__isNat(V1),V2) A__ISNATLIST(take(V1,V2)) -> A__ISNAT(V1) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U41) = 2 pi(A__U51) = 2 pi(A__U61) = 2 pi(A__ISNAT) = 1 pi(A__ISNATILIST) = 1 pi(A__ISNATLIST) = 1 Problem 1.1: SCC Processor: -> Pairs: A__U41(tt,V2) -> A__ISNATILIST(V2) A__U51(tt,V2) -> A__ISNATLIST(V2) A__U61(tt,V2) -> A__ISNATILIST(V2) A__ISNATILIST(V) -> A__ISNATLIST(V) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Reduction Pair Processor: -> Pairs: A__U71(tt,L,N) -> A__U72(a__isNat(N),L) A__U72(tt,L) -> A__LENGTH(mark(L)) A__U72(tt,L) -> MARK(L) 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) A__LENGTH(cons(N,L)) -> A__U71(a__isNatList(L),L,N) A__TAKE(s(M),cons(N,IL)) -> A__U91(a__isNatIList(IL),IL,M,N) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1,X2,X3)) -> A__U71(mark(X1),X2,X3) MARK(U71(X1,X2,X3)) -> MARK(X1) MARK(U72(X1,X2)) -> A__U72(mark(X1),X2) MARK(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> A__U91(mark(X1),X2,X3,X4) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> A__TAKE(mark(X1),mark(X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X) = X [a__U31](X) = X [a__U41](X1,X2) = X1 [a__U42](X) = X [a__U51](X1,X2) = X1 [a__U52](X) = X [a__U61](X1,X2) = X1 [a__U62](X) = X [a__U71](X1,X2,X3) = X1 + 2.X2 + X3 [a__U72](X1,X2) = X1 + 2.X2 [a__U81](X) = X [a__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__isNat](X) = 2 [a__isNatIList](X) = 2 [a__isNatList](X) = 2 [a__length](X) = 2.X + 2 [a__take](X1,X2) = 2.X1 + 2.X2 + 2 [a__zeros] = 0 [mark](X) = X [0] = 0 [U11](X) = X [U21](X) = X [U31](X) = X [U41](X1,X2) = X1 [U42](X) = X [U51](X1,X2) = X1 [U52](X) = X [U61](X1,X2) = X1 [U62](X) = X [U71](X1,X2,X3) = X1 + 2.X2 + X3 [U72](X1,X2) = X1 + 2.X2 [U81](X) = X [U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [cons](X1,X2) = 2.X1 + X2 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2.X + 2 [nil] = 2 [s](X) = X [take](X1,X2) = 2.X1 + 2.X2 + 2 [tt] = 2 [zeros] = 0 [A__U71](X1,X2,X3) = X1 + 2.X2 + 2.X3 [A__U72](X1,X2) = X1 + 2.X2 [A__U91](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 [A__U92](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 [A__U93](X1,X2,X3,X4) = 2.X3 + 2.X4 [A__LENGTH](X) = 2.X + 2 [A__TAKE](X1,X2) = 2.X1 + 2.X2 [MARK](X) = 2.X Problem 1.2: SCC Processor: -> Pairs: A__U71(tt,L,N) -> A__U72(a__isNat(N),L) A__U72(tt,L) -> A__LENGTH(mark(L)) 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) A__LENGTH(cons(N,L)) -> A__U71(a__isNatList(L),L,N) A__TAKE(s(M),cons(N,IL)) -> A__U91(a__isNatIList(IL),IL,M,N) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1,X2,X3)) -> A__U71(mark(X1),X2,X3) MARK(U71(X1,X2,X3)) -> MARK(X1) MARK(U72(X1,X2)) -> A__U72(mark(X1),X2) MARK(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> A__U91(mark(X1),X2,X3,X4) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> A__TAKE(mark(X1),mark(X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 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) ->->-> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: 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) A__TAKE(s(M),cons(N,IL)) -> A__U91(a__isNatIList(IL),IL,M,N) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1,X2,X3)) -> MARK(X1) MARK(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> A__U91(mark(X1),X2,X3,X4) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> A__TAKE(mark(X1),mark(X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros The problem is decomposed in 2 subproblems. Problem 1.2.1: Reduction Pair Processor: -> Pairs: 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) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X) = X [a__U31](X) = X + 1 [a__U41](X1,X2) = 2.X2 + 2 [a__U42](X) = X [a__U51](X1,X2) = 2.X2 + 1 [a__U52](X) = X [a__U61](X1,X2) = X1 + 2.X2 [a__U62](X) = X [a__U71](X1,X2,X3) = 2.X2 [a__U72](X1,X2) = 2.X2 [a__U81](X) = 2 [a__U91](X1,X2,X3,X4) = 2.X2 + 2.X3 [a__U92](X1,X2,X3,X4) = 2.X2 + 2.X3 [a__U93](X1,X2,X3,X4) = 2.X2 + 2.X3 [a__isNat](X) = 2.X + 1 [a__isNatIList](X) = 2.X + 2 [a__isNatList](X) = 2.X + 1 [a__length](X) = X [a__take](X1,X2) = X1 + X2 [a__zeros] = 0 [mark](X) = X [0] = 2 [U11](X) = X [U21](X) = X [U31](X) = X + 1 [U41](X1,X2) = 2.X2 + 2 [U42](X) = X [U51](X1,X2) = 2.X2 + 1 [U52](X) = X [U61](X1,X2) = X1 + 2.X2 [U62](X) = X [U71](X1,X2,X3) = 2.X2 [U72](X1,X2) = 2.X2 [U81](X) = 2 [U91](X1,X2,X3,X4) = 2.X2 + 2.X3 [U92](X1,X2,X3,X4) = 2.X2 + 2.X3 [U93](X1,X2,X3,X4) = 2.X2 + 2.X3 [cons](X1,X2) = 2.X2 [isNat](X) = 2.X + 1 [isNatIList](X) = 2.X + 2 [isNatList](X) = 2.X + 1 [length](X) = X [nil] = 2 [s](X) = 2.X [take](X1,X2) = X1 + X2 [tt] = 2 [zeros] = 0 [A__U71](X1,X2,X3) = X1 + 2.X2 + 1 [A__U72](X1,X2) = 2.X2 + 2 [A__LENGTH](X) = 2.X + 2 Problem 1.2.1: SCC Processor: -> Pairs: A__U72(tt,L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N,L)) -> A__U71(a__isNatList(L),L,N) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2.2: Reduction Pair Processor: -> Pairs: 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) A__TAKE(s(M),cons(N,IL)) -> A__U91(a__isNatIList(IL),IL,M,N) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1,X2,X3)) -> MARK(X1) MARK(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> A__U91(mark(X1),X2,X3,X4) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> A__TAKE(mark(X1),mark(X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X) = X [a__U21](X) = X [a__U31](X) = X [a__U41](X1,X2) = X1 [a__U42](X) = X [a__U51](X1,X2) = X1 [a__U52](X) = X [a__U61](X1,X2) = X1 [a__U62](X) = X [a__U71](X1,X2,X3) = X1 + 2.X2 + X3 [a__U72](X1,X2) = X1 + 2.X2 [a__U81](X) = X [a__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [a__isNat](X) = 2 [a__isNatIList](X) = 2 [a__isNatList](X) = 2 [a__length](X) = 2.X + 2 [a__take](X1,X2) = 2.X1 + 2.X2 + 2 [a__zeros] = 2 [mark](X) = X [0] = 0 [U11](X) = X [U21](X) = X [U31](X) = X [U41](X1,X2) = X1 [U42](X) = X [U51](X1,X2) = X1 [U52](X) = X [U61](X1,X2) = X1 [U62](X) = X [U71](X1,X2,X3) = X1 + 2.X2 + X3 [U72](X1,X2) = X1 + 2.X2 [U81](X) = X [U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [U92](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [U93](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [cons](X1,X2) = 2.X1 + X2 [isNat](X) = 2 [isNatIList](X) = 2 [isNatList](X) = 2 [length](X) = 2.X + 2 [nil] = 1 [s](X) = X [take](X1,X2) = 2.X1 + 2.X2 + 2 [tt] = 2 [zeros] = 2 [A__U91](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4 [A__U92](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 + 1 [A__U93](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 + 1 [A__TAKE](X1,X2) = 2.X1 + 2.X2 + 2 [MARK](X) = 2.X + 1 Problem 1.2.2: SCC Processor: -> Pairs: A__U92(tt,IL,M,N) -> A__U93(a__isNat(N),IL,M,N) A__U93(tt,IL,M,N) -> MARK(N) A__TAKE(s(M),cons(N,IL)) -> A__U91(a__isNatIList(IL),IL,M,N) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1,X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1,X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1,X2,X3)) -> MARK(X1) MARK(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> A__U91(mark(X1),X2,X3,X4) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> A__TAKE(mark(X1),mark(X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: 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(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1.2.2: Subterm Processor: -> Pairs: 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(U72(X1,X2)) -> MARK(X1) MARK(U81(X)) -> MARK(X) MARK(U91(X1,X2,X3,X4)) -> MARK(X1) MARK(U92(X1,X2,X3,X4)) -> A__U92(mark(X1),X2,X3,X4) MARK(U92(X1,X2,X3,X4)) -> MARK(X1) MARK(U93(X1,X2,X3,X4)) -> A__U93(mark(X1),X2,X3,X4) MARK(U93(X1,X2,X3,X4)) -> MARK(X1) MARK(cons(X1,X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__U92) = 4 pi(A__U93) = 4 pi(MARK) = 1 Problem 1.2.2: SCC Processor: -> Pairs: A__U92(tt,IL,M,N) -> A__U93(a__isNat(N),IL,M,N) A__U93(tt,IL,M,N) -> MARK(N) -> Rules: a__U11(tt) -> tt a__U11(X) -> U11(X) a__U21(tt) -> tt a__U21(X) -> U21(X) a__U31(tt) -> tt a__U31(X) -> U31(X) a__U41(tt,V2) -> a__U42(a__isNatIList(V2)) a__U41(X1,X2) -> U41(X1,X2) a__U42(tt) -> tt a__U42(X) -> U42(X) a__U51(tt,V2) -> a__U52(a__isNatList(V2)) a__U51(X1,X2) -> U51(X1,X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt,V2) -> a__U62(a__isNatIList(V2)) a__U61(X1,X2) -> U61(X1,X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt,L,N) -> a__U72(a__isNat(N),L) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(tt,L) -> s(a__length(mark(L))) a__U72(X1,X2) -> U72(X1,X2) a__U81(tt) -> nil a__U81(X) -> U81(X) a__U91(tt,IL,M,N) -> a__U92(a__isNat(M),IL,M,N) a__U91(X1,X2,X3,X4) -> U91(X1,X2,X3,X4) a__U92(tt,IL,M,N) -> a__U93(a__isNat(N),IL,M,N) a__U92(X1,X2,X3,X4) -> U92(X1,X2,X3,X4) a__U93(tt,IL,M,N) -> cons(mark(N),take(M,IL)) a__U93(X1,X2,X3,X4) -> U93(X1,X2,X3,X4) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatList(V1)) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(X) -> isNat(X) a__isNatIList(cons(V1,V2)) -> a__U41(a__isNat(V1),V2) a__isNatIList(zeros) -> tt a__isNatIList(V) -> a__U31(a__isNatList(V)) a__isNatIList(X) -> isNatIList(X) a__isNatList(cons(V1,V2)) -> a__U51(a__isNat(V1),V2) a__isNatList(nil) -> tt a__isNatList(take(V1,V2)) -> a__U61(a__isNat(V1),V2) a__isNatList(X) -> isNatList(X) a__length(cons(N,L)) -> a__U71(a__isNatList(L),L,N) a__length(nil) -> 0 a__length(X) -> length(X) a__take(0,IL) -> a__U81(a__isNatIList(IL)) a__take(s(M),cons(N,IL)) -> a__U91(a__isNatIList(IL),IL,M,N) a__take(X1,X2) -> take(X1,X2) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 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(U51(X1,X2)) -> a__U51(mark(X1),X2) mark(U52(X)) -> a__U52(mark(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(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(cons(X1,X2)) -> cons(mark(X1),X2) mark(isNat(X)) -> a__isNat(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(length(X)) -> a__length(mark(X)) mark(nil) -> nil mark(s(X)) -> s(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite.