/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR IL L M N V V1 V2 X X1 X2 X3 X4)
(RULES
a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
a__U101(X1,X2,X3) -> U101(X1,X2,X3)
a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
a__U102(X1,X2,X3) -> U102(X1,X2,X3)
a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
a__U103(X1,X2,X3) -> U103(X1,X2,X3)
a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
a__U104(X1,X2,X3) -> U104(X1,X2,X3)
a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
a__U105(X1,X2) -> U105(X1,X2)
a__U106(tt) -> tt
a__U106(X) -> U106(X)
a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
a__U11(X1,X2) -> U11(X1,X2)
a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
a__U111(X1,X2,X3) -> U111(X1,X2,X3)
a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
a__U112(X1,X2,X3) -> U112(X1,X2,X3)
a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
a__U113(X1,X2,X3) -> U113(X1,X2,X3)
a__U114(tt,L) -> s(a__length(mark(L)))
a__U114(X1,X2) -> U114(X1,X2)
a__U12(tt,V1) -> a__U13(a__isNatList(V1))
a__U12(X1,X2) -> U12(X1,X2)
a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
a__U121(X1,X2) -> U121(X1,X2)
a__U122(tt) -> nil
a__U122(X) -> U122(X)
a__U13(tt) -> tt
a__U13(X) -> U13(X)
a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(tt,V1) -> a__U23(a__isNat(V1))
a__U22(X1,X2) -> U22(X1,X2)
a__U23(tt) -> tt
a__U23(X) -> U23(X)
a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(tt,V) -> a__U33(a__isNatList(V))
a__U32(X1,X2) -> U32(X1,X2)
a__U33(tt) -> tt
a__U33(X) -> U33(X)
a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
a__U41(X1,X2,X3) -> U41(X1,X2,X3)
a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
a__U42(X1,X2,X3) -> U42(X1,X2,X3)
a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
a__U43(X1,X2,X3) -> U43(X1,X2,X3)
a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
a__U44(X1,X2,X3) -> U44(X1,X2,X3)
a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
a__U45(X1,X2) -> U45(X1,X2)
a__U46(tt) -> tt
a__U46(X) -> U46(X)
a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
a__U51(X1,X2) -> U51(X1,X2)
a__U52(tt) -> tt
a__U52(X) -> U52(X)
a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
a__U61(X1,X2) -> U61(X1,X2)
a__U62(tt) -> tt
a__U62(X) -> U62(X)
a__U71(tt) -> tt
a__U71(X) -> U71(X)
a__U81(tt) -> tt
a__U81(X) -> U81(X)
a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
a__U91(X1,X2,X3) -> U91(X1,X2,X3)
a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
a__U92(X1,X2,X3) -> U92(X1,X2,X3)
a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
a__U93(X1,X2,X3) -> U93(X1,X2,X3)
a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
a__U94(X1,X2,X3) -> U94(X1,X2,X3)
a__U95(tt,V2) -> a__U96(a__isNatList(V2))
a__U95(X1,X2) -> U95(X1,X2)
a__U96(tt) -> tt
a__U96(X) -> U96(X)
a__isNat(0) -> tt
a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNat(X) -> isNat(X)
a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
a__isNatIList(zeros) -> tt
a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
a__isNatIList(X) -> isNatIList(X)
a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
a__isNatIListKind(nil) -> tt
a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
a__isNatIListKind(zeros) -> tt
a__isNatIListKind(X) -> isNatIListKind(X)
a__isNatKind(0) -> tt
a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
a__isNatKind(X) -> isNatKind(X)
a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
a__isNatList(nil) -> tt
a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
a__isNatList(X) -> isNatList(X)
a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
a__length(nil) -> 0
a__length(X) -> length(X)
a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
mark(U106(X)) -> a__U106(mark(X))
mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
mark(U122(X)) -> a__U122(mark(X))
mark(U13(X)) -> a__U13(mark(X))
mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
mark(U33(X)) -> a__U33(mark(X))
mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
mark(U81(X)) -> a__U81(mark(X))
mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
mark(U96(X)) -> a__U96(mark(X))
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(isNatIList(X)) -> a__isNatIList(X)
mark(isNatIListKind(X)) -> a__isNatIListKind(X)
mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> A__U102(a__isNatKind(V1),V1,V2)
 A__U101(tt,V1,V2) -> A__ISNATKIND(V1)
 A__U102(tt,V1,V2) -> A__U103(a__isNatIListKind(V2),V1,V2)
 A__U102(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U103(tt,V1,V2) -> A__U104(a__isNatIListKind(V2),V1,V2)
 A__U103(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U104(tt,V1,V2) -> A__U105(a__isNat(V1),V2)
 A__U104(tt,V1,V2) -> A__ISNAT(V1)
 A__U105(tt,V2) -> A__U106(a__isNatIList(V2))
 A__U105(tt,V2) -> A__ISNATILIST(V2)
 A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1)
 A__U11(tt,V1) -> A__ISNATILISTKIND(V1)
 A__U111(tt,L,N) -> A__U112(a__isNatIListKind(L),L,N)
 A__U111(tt,L,N) -> A__ISNATILISTKIND(L)
 A__U112(tt,L,N) -> A__U113(a__isNat(N),L,N)
 A__U112(tt,L,N) -> A__ISNAT(N)
 A__U113(tt,L,N) -> A__U114(a__isNatKind(N),L)
 A__U113(tt,L,N) -> A__ISNATKIND(N)
 A__U114(tt,L) -> A__LENGTH(mark(L))
 A__U114(tt,L) -> MARK(L)
 A__U12(tt,V1) -> A__U13(a__isNatList(V1))
 A__U12(tt,V1) -> A__ISNATLIST(V1)
 A__U121(tt,IL) -> A__U122(a__isNatIListKind(IL))
 A__U121(tt,IL) -> A__ISNATILISTKIND(IL)
 A__U131(tt,IL,M,N) -> A__U132(a__isNatIListKind(IL),IL,M,N)
 A__U131(tt,IL,M,N) -> A__ISNATILISTKIND(IL)
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U132(tt,IL,M,N) -> A__ISNAT(M)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__ISNATKIND(M)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U134(tt,IL,M,N) -> A__ISNAT(N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__ISNATKIND(N)
 A__U136(tt,IL,M,N) -> MARK(N)
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U21(tt,V1) -> A__ISNATKIND(V1)
 A__U22(tt,V1) -> A__U23(a__isNat(V1))
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__U31(tt,V) -> A__U32(a__isNatIListKind(V),V)
 A__U31(tt,V) -> A__ISNATILISTKIND(V)
 A__U32(tt,V) -> A__U33(a__isNatList(V))
 A__U32(tt,V) -> A__ISNATLIST(V)
 A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2)
 A__U41(tt,V1,V2) -> A__ISNATKIND(V1)
 A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2)
 A__U42(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2)
 A__U43(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2)
 A__U44(tt,V1,V2) -> A__ISNAT(V1)
 A__U45(tt,V2) -> A__U46(a__isNatIList(V2))
 A__U45(tt,V2) -> A__ISNATILIST(V2)
 A__U51(tt,V2) -> A__U52(a__isNatIListKind(V2))
 A__U51(tt,V2) -> A__ISNATILISTKIND(V2)
 A__U61(tt,V2) -> A__U62(a__isNatIListKind(V2))
 A__U61(tt,V2) -> A__ISNATILISTKIND(V2)
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U91(tt,V1,V2) -> A__ISNATKIND(V1)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U92(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U94(tt,V1,V2) -> A__ISNAT(V1)
 A__U95(tt,V2) -> A__U96(a__isNatList(V2))
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1)
 A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
 A__ISNAT(s(V1)) -> A__ISNATKIND(V1)
 A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2)
 A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V),V)
 A__ISNATILIST(V) -> A__ISNATILISTKIND(V)
 A__ISNATILISTKIND(cons(V1,V2)) -> A__U51(a__isNatKind(V1),V2)
 A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATILISTKIND(take(V1,V2)) -> A__U61(a__isNatKind(V1),V2)
 A__ISNATILISTKIND(take(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATKIND(length(V1)) -> A__U71(a__isNatIListKind(V1))
 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1)
 A__ISNATKIND(s(V1)) -> A__U81(a__isNatKind(V1))
 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
 A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATLIST(take(V1,V2)) -> A__U101(a__isNatKind(V1),V1,V2)
 A__ISNATLIST(take(V1,V2)) -> A__ISNATKIND(V1)
 A__LENGTH(cons(N,L)) -> A__U111(a__isNatList(L),L,N)
 A__LENGTH(cons(N,L)) -> A__ISNATLIST(L)
 A__TAKE(0,IL) -> A__U121(a__isNatIList(IL),IL)
 A__TAKE(0,IL) -> A__ISNATILIST(IL)
 A__TAKE(s(M),cons(N,IL)) -> A__U131(a__isNatIList(IL),IL,M,N)
 A__TAKE(s(M),cons(N,IL)) -> A__ISNATILIST(IL)
 MARK(U101(X1,X2,X3)) -> A__U101(mark(X1),X2,X3)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> A__U102(mark(X1),X2,X3)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> A__U103(mark(X1),X2,X3)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> A__U104(mark(X1),X2,X3)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> A__U105(mark(X1),X2)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> A__U106(mark(X))
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> A__U11(mark(X1),X2)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> A__U111(mark(X1),X2,X3)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> A__U112(mark(X1),X2,X3)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> A__U113(mark(X1),X2,X3)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> A__U114(mark(X1),X2)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> A__U12(mark(X1),X2)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> A__U121(mark(X1),X2)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> A__U122(mark(X))
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> A__U13(mark(X))
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> A__U131(mark(X1),X2,X3,X4)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> A__U21(mark(X1),X2)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> A__U22(mark(X1),X2)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> A__U23(mark(X))
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> A__U31(mark(X1),X2)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> A__U32(mark(X1),X2)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> A__U33(mark(X))
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> A__U42(mark(X1),X2,X3)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> A__U43(mark(X1),X2,X3)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> A__U44(mark(X1),X2,X3)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> A__U45(mark(X1),X2)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(X)) -> A__U46(mark(X))
 MARK(U46(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(X)) -> A__U71(mark(X))
 MARK(U71(X)) -> MARK(X)
 MARK(U81(X)) -> A__U81(mark(X))
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> A__U91(mark(X1),X2,X3)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> A__U94(mark(X1),X2,X3)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> A__U95(mark(X1),X2)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> A__U96(mark(X))
 MARK(U96(X)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(isNat(X)) -> A__ISNAT(X)
 MARK(isNatIList(X)) -> A__ISNATILIST(X)
 MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X)
 MARK(isNatKind(X)) -> A__ISNATKIND(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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> A__U102(a__isNatKind(V1),V1,V2)
 A__U101(tt,V1,V2) -> A__ISNATKIND(V1)
 A__U102(tt,V1,V2) -> A__U103(a__isNatIListKind(V2),V1,V2)
 A__U102(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U103(tt,V1,V2) -> A__U104(a__isNatIListKind(V2),V1,V2)
 A__U103(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U104(tt,V1,V2) -> A__U105(a__isNat(V1),V2)
 A__U104(tt,V1,V2) -> A__ISNAT(V1)
 A__U105(tt,V2) -> A__U106(a__isNatIList(V2))
 A__U105(tt,V2) -> A__ISNATILIST(V2)
 A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1)
 A__U11(tt,V1) -> A__ISNATILISTKIND(V1)
 A__U111(tt,L,N) -> A__U112(a__isNatIListKind(L),L,N)
 A__U111(tt,L,N) -> A__ISNATILISTKIND(L)
 A__U112(tt,L,N) -> A__U113(a__isNat(N),L,N)
 A__U112(tt,L,N) -> A__ISNAT(N)
 A__U113(tt,L,N) -> A__U114(a__isNatKind(N),L)
 A__U113(tt,L,N) -> A__ISNATKIND(N)
 A__U114(tt,L) -> A__LENGTH(mark(L))
 A__U114(tt,L) -> MARK(L)
 A__U12(tt,V1) -> A__U13(a__isNatList(V1))
 A__U12(tt,V1) -> A__ISNATLIST(V1)
 A__U121(tt,IL) -> A__U122(a__isNatIListKind(IL))
 A__U121(tt,IL) -> A__ISNATILISTKIND(IL)
 A__U131(tt,IL,M,N) -> A__U132(a__isNatIListKind(IL),IL,M,N)
 A__U131(tt,IL,M,N) -> A__ISNATILISTKIND(IL)
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U132(tt,IL,M,N) -> A__ISNAT(M)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__ISNATKIND(M)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U134(tt,IL,M,N) -> A__ISNAT(N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__ISNATKIND(N)
 A__U136(tt,IL,M,N) -> MARK(N)
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U21(tt,V1) -> A__ISNATKIND(V1)
 A__U22(tt,V1) -> A__U23(a__isNat(V1))
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__U31(tt,V) -> A__U32(a__isNatIListKind(V),V)
 A__U31(tt,V) -> A__ISNATILISTKIND(V)
 A__U32(tt,V) -> A__U33(a__isNatList(V))
 A__U32(tt,V) -> A__ISNATLIST(V)
 A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2)
 A__U41(tt,V1,V2) -> A__ISNATKIND(V1)
 A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2)
 A__U42(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2)
 A__U43(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2)
 A__U44(tt,V1,V2) -> A__ISNAT(V1)
 A__U45(tt,V2) -> A__U46(a__isNatIList(V2))
 A__U45(tt,V2) -> A__ISNATILIST(V2)
 A__U51(tt,V2) -> A__U52(a__isNatIListKind(V2))
 A__U51(tt,V2) -> A__ISNATILISTKIND(V2)
 A__U61(tt,V2) -> A__U62(a__isNatIListKind(V2))
 A__U61(tt,V2) -> A__ISNATILISTKIND(V2)
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U91(tt,V1,V2) -> A__ISNATKIND(V1)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U92(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__ISNATILISTKIND(V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U94(tt,V1,V2) -> A__ISNAT(V1)
 A__U95(tt,V2) -> A__U96(a__isNatList(V2))
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1)
 A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
 A__ISNAT(s(V1)) -> A__ISNATKIND(V1)
 A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2)
 A__ISNATILIST(cons(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V),V)
 A__ISNATILIST(V) -> A__ISNATILISTKIND(V)
 A__ISNATILISTKIND(cons(V1,V2)) -> A__U51(a__isNatKind(V1),V2)
 A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATILISTKIND(take(V1,V2)) -> A__U61(a__isNatKind(V1),V2)
 A__ISNATILISTKIND(take(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATKIND(length(V1)) -> A__U71(a__isNatIListKind(V1))
 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1)
 A__ISNATKIND(s(V1)) -> A__U81(a__isNatKind(V1))
 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
 A__ISNATLIST(cons(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATLIST(take(V1,V2)) -> A__U101(a__isNatKind(V1),V1,V2)
 A__ISNATLIST(take(V1,V2)) -> A__ISNATKIND(V1)
 A__LENGTH(cons(N,L)) -> A__U111(a__isNatList(L),L,N)
 A__LENGTH(cons(N,L)) -> A__ISNATLIST(L)
 A__TAKE(0,IL) -> A__U121(a__isNatIList(IL),IL)
 A__TAKE(0,IL) -> A__ISNATILIST(IL)
 A__TAKE(s(M),cons(N,IL)) -> A__U131(a__isNatIList(IL),IL,M,N)
 A__TAKE(s(M),cons(N,IL)) -> A__ISNATILIST(IL)
 MARK(U101(X1,X2,X3)) -> A__U101(mark(X1),X2,X3)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> A__U102(mark(X1),X2,X3)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> A__U103(mark(X1),X2,X3)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> A__U104(mark(X1),X2,X3)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> A__U105(mark(X1),X2)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> A__U106(mark(X))
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> A__U11(mark(X1),X2)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> A__U111(mark(X1),X2,X3)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> A__U112(mark(X1),X2,X3)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> A__U113(mark(X1),X2,X3)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> A__U114(mark(X1),X2)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> A__U12(mark(X1),X2)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> A__U121(mark(X1),X2)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> A__U122(mark(X))
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> A__U13(mark(X))
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> A__U131(mark(X1),X2,X3,X4)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> A__U21(mark(X1),X2)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> A__U22(mark(X1),X2)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> A__U23(mark(X))
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> A__U31(mark(X1),X2)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> A__U32(mark(X1),X2)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> A__U33(mark(X))
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> A__U41(mark(X1),X2,X3)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> A__U42(mark(X1),X2,X3)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> A__U43(mark(X1),X2,X3)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> A__U44(mark(X1),X2,X3)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> A__U45(mark(X1),X2)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(X)) -> A__U46(mark(X))
 MARK(U46(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(X)) -> A__U71(mark(X))
 MARK(U71(X)) -> MARK(X)
 MARK(U81(X)) -> A__U81(mark(X))
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> A__U91(mark(X1),X2,X3)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> A__U92(mark(X1),X2,X3)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> A__U93(mark(X1),X2,X3)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> A__U94(mark(X1),X2,X3)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> A__U95(mark(X1),X2)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> A__U96(mark(X))
 MARK(U96(X)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(isNat(X)) -> A__ISNAT(X)
 MARK(isNatIList(X)) -> A__ISNATILIST(X)
 MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X)
 MARK(isNatKind(X)) -> A__ISNATKIND(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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U51(tt,V2) -> A__ISNATILISTKIND(V2)
 A__U61(tt,V2) -> A__ISNATILISTKIND(V2)
 A__ISNATILISTKIND(cons(V1,V2)) -> A__U51(a__isNatKind(V1),V2)
 A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATILISTKIND(take(V1,V2)) -> A__U61(a__isNatKind(V1),V2)
 A__ISNATILISTKIND(take(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1)
 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1)
->->-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> A__U102(a__isNatKind(V1),V1,V2)
 A__U102(tt,V1,V2) -> A__U103(a__isNatIListKind(V2),V1,V2)
 A__U103(tt,V1,V2) -> A__U104(a__isNatIListKind(V2),V1,V2)
 A__U104(tt,V1,V2) -> A__U105(a__isNat(V1),V2)
 A__U104(tt,V1,V2) -> A__ISNAT(V1)
 A__U105(tt,V2) -> A__ISNATILIST(V2)
 A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1)
 A__U12(tt,V1) -> A__ISNATLIST(V1)
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__U31(tt,V) -> A__U32(a__isNatIListKind(V),V)
 A__U32(tt,V) -> A__ISNATLIST(V)
 A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2)
 A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2)
 A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2)
 A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2)
 A__U44(tt,V1,V2) -> A__ISNAT(V1)
 A__U45(tt,V2) -> A__ISNATILIST(V2)
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U94(tt,V1,V2) -> A__ISNAT(V1)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
 A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2)
 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V),V)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
 A__ISNATLIST(take(V1,V2)) -> A__U101(a__isNatKind(V1),V1,V2)
->->-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U111(tt,L,N) -> A__U112(a__isNatIListKind(L),L,N)
 A__U112(tt,L,N) -> A__U113(a__isNat(N),L,N)
 A__U113(tt,L,N) -> A__U114(a__isNatKind(N),L)
 A__U114(tt,L) -> A__LENGTH(mark(L))
 A__U114(tt,L) -> MARK(L)
 A__U131(tt,IL,M,N) -> A__U132(a__isNatIListKind(IL),IL,M,N)
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
 A__LENGTH(cons(N,L)) -> A__U111(a__isNatList(L),L,N)
 A__TAKE(s(M),cons(N,IL)) -> A__U131(a__isNatIList(IL),IL,M,N)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> A__U111(mark(X1),X2,X3)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> A__U112(mark(X1),X2,X3)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> A__U113(mark(X1),X2,X3)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> A__U114(mark(X1),X2)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> A__U131(mark(X1),X2,X3,X4)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(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(X)) -> MARK(X)
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> MARK(X)
 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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 A__U51(tt,V2) -> A__ISNATILISTKIND(V2)
 A__U61(tt,V2) -> A__ISNATILISTKIND(V2)
 A__ISNATILISTKIND(cons(V1,V2)) -> A__U51(a__isNatKind(V1),V2)
 A__ISNATILISTKIND(cons(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATILISTKIND(take(V1,V2)) -> A__U61(a__isNatKind(V1),V2)
 A__ISNATILISTKIND(take(V1,V2)) -> A__ISNATKIND(V1)
 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1)
 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U51) = 2
 pi(A__U61) = 2
 pi(A__ISNATILISTKIND) = 1
 pi(A__ISNATKIND) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 A__U51(tt,V2) -> A__ISNATILISTKIND(V2)
 A__U61(tt,V2) -> A__ISNATILISTKIND(V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> A__U102(a__isNatKind(V1),V1,V2)
 A__U102(tt,V1,V2) -> A__U103(a__isNatIListKind(V2),V1,V2)
 A__U103(tt,V1,V2) -> A__U104(a__isNatIListKind(V2),V1,V2)
 A__U104(tt,V1,V2) -> A__U105(a__isNat(V1),V2)
 A__U104(tt,V1,V2) -> A__ISNAT(V1)
 A__U105(tt,V2) -> A__ISNATILIST(V2)
 A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1)
 A__U12(tt,V1) -> A__ISNATLIST(V1)
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__U31(tt,V) -> A__U32(a__isNatIListKind(V),V)
 A__U32(tt,V) -> A__ISNATLIST(V)
 A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2)
 A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2)
 A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2)
 A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2)
 A__U44(tt,V1,V2) -> A__ISNAT(V1)
 A__U45(tt,V2) -> A__ISNATILIST(V2)
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U94(tt,V1,V2) -> A__ISNAT(V1)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
 A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2)
 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V),V)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
 A__ISNATLIST(take(V1,V2)) -> A__U101(a__isNatKind(V1),V1,V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__U101](X1,X2,X3) = 2.X3 + 2
[a__U102](X1,X2,X3) = 2.X3 + 2
[a__U103](X1,X2,X3) = 2.X3 + 2
[a__U104](X1,X2,X3) = X1 + 2
[a__U105](X1,X2) = 2
[a__U106](X) = 2
[a__U11](X1,X2) = 2.X2 + 2
[a__U12](X1,X2) = X1 + 2
[a__U13](X) = 2
[a__U21](X1,X2) = 2.X2 + 2
[a__U22](X1,X2) = X1 + X2 + 2
[a__U23](X) = X + 2
[a__U31](X1,X2) = X1 + 2
[a__U32](X1,X2) = 2
[a__U33](X) = 2
[a__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[a__U42](X1,X2,X3) = 2.X2 + 2.X3 + 2
[a__U43](X1,X2,X3) = 2.X2 + 2.X3 + 2
[a__U44](X1,X2,X3) = X1 + 2
[a__U45](X1,X2) = 2
[a__U46](X) = 2
[a__U51](X1,X2) = X1 + 2.X2 + 2
[a__U52](X) = X + 1
[a__U61](X1,X2) = 2.X1 + 2.X2 + 1
[a__U62](X) = X + 1
[a__U71](X) = X + 2
[a__U81](X) = 2.X + 1
[a__U91](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[a__U92](X1,X2,X3) = 2.X1 + 2.X3 + 2
[a__U93](X1,X2,X3) = 2.X3 + 2
[a__U94](X1,X2,X3) = X1 + 2
[a__U95](X1,X2) = 2
[a__U96](X) = 2
[a__isNat](X) = X + 2
[a__isNatIList](X) = 2.X + 2
[a__isNatIListKind](X) = 2.X
[a__isNatKind](X) = X
[a__isNatList](X) = 2.X
[0] = 2
[U101](X1,X2,X3) = 2.X3 + 1
[U102](X1,X2,X3) = 2.X3 + 1
[U103](X1,X2,X3) = 2.X3 + 2
[U104](X1,X2,X3) = 0
[U105](X1,X2) = 0
[U106](X) = 0
[U11](X1,X2) = 2.X2 + 1
[U12](X1,X2) = X1 + 2
[U13](X) = 1
[U21](X1,X2) = 2
[U22](X1,X2) = X1 + 2
[U23](X) = X + 2
[U31](X1,X2) = 0
[U32](X1,X2) = 1
[U33](X) = 2
[U41](X1,X2,X3) = X1 + 2.X2 + X3 + 2
[U42](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U43](X1,X2,X3) = 2.X2 + X3 + 2
[U44](X1,X2,X3) = X1 + 2
[U45](X1,X2) = 0
[U46](X) = 2
[U51](X1,X2) = X1 + 1
[U52](X) = X + 1
[U61](X1,X2) = 2.X1 + 2.X2 + 1
[U62](X) = X + 1
[U71](X) = 2
[U81](X) = 2.X + 1
[U91](X1,X2,X3) = X1 + 2.X2 + X3
[U92](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U93](X1,X2,X3) = 2.X3 + 2
[U94](X1,X2,X3) = X1 + 2
[U95](X1,X2) = 2
[U96](X) = 2
[cons](X1,X2) = 2.X1 + 2.X2 + 2
[isNat](X) = X + 2
[isNatIList](X) = X + 2
[isNatIListKind](X) = 2.X
[isNatKind](X) = X
[isNatList](X) = 2.X
[length](X) = 2.X + 2
[nil] = 1
[s](X) = 2.X + 2
[take](X1,X2) = 2.X1 + 2.X2 + 2
[tt] = 2
[zeros] = 2
[A__U101](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[A__U102](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U103](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U104](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U105](X1,X2) = 2.X2 + 1
[A__U11](X1,X2) = X1 + 2.X2 + 2
[A__U12](X1,X2) = 2.X2
[A__U21](X1,X2) = 2.X1 + 2.X2 + 2
[A__U22](X1,X2) = 2.X2 + 2
[A__U31](X1,X2) = 2.X2
[A__U32](X1,X2) = 2.X2
[A__U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1
[A__U42](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U43](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U44](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U45](X1,X2) = 2.X2 + 2
[A__U91](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[A__U92](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U93](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U94](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U95](X1,X2) = 2.X2 + 1
[A__ISNAT](X) = 2.X + 2
[A__ISNATILIST](X) = 2.X
[A__ISNATLIST](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 A__U102(tt,V1,V2) -> A__U103(a__isNatIListKind(V2),V1,V2)
 A__U103(tt,V1,V2) -> A__U104(a__isNatIListKind(V2),V1,V2)
 A__U104(tt,V1,V2) -> A__U105(a__isNat(V1),V2)
 A__U104(tt,V1,V2) -> A__ISNAT(V1)
 A__U105(tt,V2) -> A__ISNATILIST(V2)
 A__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1)
 A__U12(tt,V1) -> A__ISNATLIST(V1)
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__U31(tt,V) -> A__U32(a__isNatIListKind(V),V)
 A__U32(tt,V) -> A__ISNATLIST(V)
 A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2)
 A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2)
 A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2)
 A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2)
 A__U44(tt,V1,V2) -> A__ISNAT(V1)
 A__U45(tt,V2) -> A__ISNATILIST(V2)
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U94(tt,V1,V2) -> A__ISNAT(V1)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
 A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2)
 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V),V)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
 A__ISNATLIST(take(V1,V2)) -> A__U101(a__isNatKind(V1),V1,V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1)
 A__U12(tt,V1) -> A__ISNATLIST(V1)
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U94(tt,V1,V2) -> A__ISNAT(V1)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
->->-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2)
 A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2)
 A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2)
 A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2)
 A__U45(tt,V2) -> A__ISNATILIST(V2)
 A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2)
->->-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U11(tt,V1) -> A__U12(a__isNatIListKind(V1),V1)
 A__U12(tt,V1) -> A__ISNATLIST(V1)
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U94(tt,V1,V2) -> A__ISNAT(V1)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__U101](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[a__U102](X1,X2,X3) = X1 + X2 + 2.X3 + 2
[a__U103](X1,X2,X3) = 2.X1 + X2 + 2
[a__U104](X1,X2,X3) = X2 + 2
[a__U105](X1,X2) = X1 + 2
[a__U106](X) = 2
[a__U11](X1,X2) = 2.X2 + 2
[a__U12](X1,X2) = 2.X2 + 2
[a__U13](X) = X
[a__U21](X1,X2) = 2.X2 + 2
[a__U22](X1,X2) = 2.X2 + 2
[a__U23](X) = 2.X + 2
[a__U31](X1,X2) = 2.X2 + 2
[a__U32](X1,X2) = 2.X1
[a__U33](X) = 2
[a__U41](X1,X2,X3) = X1 + 2.X2 + 2.X3
[a__U42](X1,X2,X3) = 2.X2 + 2.X3 + 2
[a__U43](X1,X2,X3) = X1 + 2.X2 + X3 + 1
[a__U44](X1,X2,X3) = 2.X2 + X3 + 2
[a__U45](X1,X2) = 2.X1 + X2
[a__U46](X) = 2
[a__U51](X1,X2) = 2.X2 + 2
[a__U52](X) = 2.X
[a__U61](X1,X2) = 2.X1 + 2.X2
[a__U62](X) = 2.X + 2
[a__U71](X) = 2
[a__U81](X) = 2.X + 2
[a__U91](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[a__U92](X1,X2,X3) = 2.X1 + 2.X3 + 2
[a__U93](X1,X2,X3) = 2.X1 + 2
[a__U94](X1,X2,X3) = 2
[a__U95](X1,X2) = 2
[a__U96](X) = 2
[a__isNat](X) = X
[a__isNatIList](X) = 2.X + 2
[a__isNatIListKind](X) = X + 1
[a__isNatKind](X) = X
[a__isNatList](X) = 2.X + 2
[0] = 2
[U101](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
[U102](X1,X2,X3) = X1 + X2 + 2.X3 + 2
[U103](X1,X2,X3) = 2.X1 + 2
[U104](X1,X2,X3) = 2
[U105](X1,X2) = X1 + 2
[U106](X) = 2
[U11](X1,X2) = 2.X2 + 1
[U12](X1,X2) = 1
[U13](X) = X
[U21](X1,X2) = X2
[U22](X1,X2) = X2 + 1
[U23](X) = 2.X
[U31](X1,X2) = 1
[U32](X1,X2) = 2.X1
[U33](X) = 2
[U41](X1,X2,X3) = X1 + 2.X2 + X3
[U42](X1,X2,X3) = 2.X2 + 2
[U43](X1,X2,X3) = X1 + X2 + X3 + 1
[U44](X1,X2,X3) = 2.X2 + X3 + 1
[U45](X1,X2) = X1 + X2
[U46](X) = 2
[U51](X1,X2) = 2.X2 + 2
[U52](X) = X
[U61](X1,X2) = 2.X1 + X2
[U62](X) = 2.X
[U71](X) = 0
[U81](X) = X + 2
[U91](X1,X2,X3) = 2.X1 + 2.X2 + 1
[U92](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U93](X1,X2,X3) = X1 + 2
[U94](X1,X2,X3) = 2
[U95](X1,X2) = 2
[U96](X) = 1
[cons](X1,X2) = 2.X1 + 2.X2 + 2
[isNat](X) = X
[isNatIList](X) = 2.X + 1
[isNatIListKind](X) = X + 1
[isNatKind](X) = 0
[isNatList](X) = X
[length](X) = 2.X + 2
[nil] = 1
[s](X) = 2.X + 2
[take](X1,X2) = 2.X1 + 2.X2 + 2
[tt] = 2
[zeros] = 1
[A__U11](X1,X2) = 2.X1 + 2.X2 + 1
[A__U12](X1,X2) = X1 + X2 + 2
[A__U21](X1,X2) = 2.X1 + 2.X2 + 2
[A__U22](X1,X2) = 2.X2 + 2
[A__U91](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U92](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U93](X1,X2,X3) = 2.X2 + 2.X3 + 2
[A__U94](X1,X2,X3) = X1 + 2.X2 + X3 + 1
[A__U95](X1,X2) = 2.X1 + X2 + 2
[A__ISNAT](X) = 2.X + 2
[A__ISNATLIST](X) = X + 2

Problem 1.2.1: 

SCC Processor:
-> Pairs:
 A__U12(tt,V1) -> A__ISNATLIST(V1)
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U94(tt,V1,V2) -> A__ISNAT(V1)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1),V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
->->-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
->->-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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 1.2.1.1: 

Subterm Processor:
-> Pairs:
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1),V1)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U21) = 2
 pi(A__U22) = 2
 pi(A__ISNAT) = 1

Problem 1.2.1.1: 

SCC Processor:
-> Pairs:
 A__U21(tt,V1) -> A__U22(a__isNatKind(V1),V1)
 A__U22(tt,V1) -> A__ISNAT(V1)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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 1.2.1.2: 

Subterm Processor:
-> Pairs:
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
 A__ISNATLIST(cons(V1,V2)) -> A__U91(a__isNatKind(V1),V1,V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U91) = 3
 pi(A__U92) = 3
 pi(A__U93) = 3
 pi(A__U94) = 3
 pi(A__U95) = 2
 pi(A__ISNATLIST) = 1

Problem 1.2.1.2: 

SCC Processor:
-> Pairs:
 A__U91(tt,V1,V2) -> A__U92(a__isNatKind(V1),V1,V2)
 A__U92(tt,V1,V2) -> A__U93(a__isNatIListKind(V2),V1,V2)
 A__U93(tt,V1,V2) -> A__U94(a__isNatIListKind(V2),V1,V2)
 A__U94(tt,V1,V2) -> A__U95(a__isNat(V1),V2)
 A__U95(tt,V2) -> A__ISNATLIST(V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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: 

Subterm Processor:
-> Pairs:
 A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2)
 A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2)
 A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2)
 A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2)
 A__U45(tt,V2) -> A__ISNATILIST(V2)
 A__ISNATILIST(cons(V1,V2)) -> A__U41(a__isNatKind(V1),V1,V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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) = 3
 pi(A__U42) = 3
 pi(A__U43) = 3
 pi(A__U44) = 3
 pi(A__U45) = 2
 pi(A__ISNATILIST) = 1

Problem 1.2.2: 

SCC Processor:
-> Pairs:
 A__U41(tt,V1,V2) -> A__U42(a__isNatKind(V1),V1,V2)
 A__U42(tt,V1,V2) -> A__U43(a__isNatIListKind(V2),V1,V2)
 A__U43(tt,V1,V2) -> A__U44(a__isNatIListKind(V2),V1,V2)
 A__U44(tt,V1,V2) -> A__U45(a__isNat(V1),V2)
 A__U45(tt,V2) -> A__ISNATILIST(V2)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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 1.3: 

Reduction Pair Processor:
-> Pairs:
 A__U111(tt,L,N) -> A__U112(a__isNatIListKind(L),L,N)
 A__U112(tt,L,N) -> A__U113(a__isNat(N),L,N)
 A__U113(tt,L,N) -> A__U114(a__isNatKind(N),L)
 A__U114(tt,L) -> A__LENGTH(mark(L))
 A__U114(tt,L) -> MARK(L)
 A__U131(tt,IL,M,N) -> A__U132(a__isNatIListKind(IL),IL,M,N)
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
 A__LENGTH(cons(N,L)) -> A__U111(a__isNatList(L),L,N)
 A__TAKE(s(M),cons(N,IL)) -> A__U131(a__isNatIList(IL),IL,M,N)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> A__U111(mark(X1),X2,X3)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> A__U112(mark(X1),X2,X3)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> A__U113(mark(X1),X2,X3)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> A__U114(mark(X1),X2)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> A__U131(mark(X1),X2,X3,X4)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(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(X)) -> MARK(X)
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> MARK(X)
 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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101](X1,X2,X3) = X1
[a__U102](X1,X2,X3) = X1
[a__U103](X1,X2,X3) = X1
[a__U104](X1,X2,X3) = X1
[a__U105](X1,X2) = X1
[a__U106](X) = X
[a__U11](X1,X2) = X1
[a__U111](X1,X2,X3) = X1 + X2 + 2.X3
[a__U112](X1,X2,X3) = X1 + X2 + 2.X3
[a__U113](X1,X2,X3) = X1 + X2
[a__U114](X1,X2) = X1 + X2
[a__U12](X1,X2) = X1
[a__U121](X1,X2) = X1 + 2.X2
[a__U122](X) = X
[a__U13](X) = X
[a__U131](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U132](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U133](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U134](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U135](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U136](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U21](X1,X2) = X1
[a__U22](X1,X2) = X1
[a__U23](X) = X
[a__U31](X1,X2) = X1
[a__U32](X1,X2) = X1
[a__U33](X) = X
[a__U41](X1,X2,X3) = X1
[a__U42](X1,X2,X3) = X1
[a__U43](X1,X2,X3) = X1
[a__U44](X1,X2,X3) = X1
[a__U45](X1,X2) = X1
[a__U46](X) = X
[a__U51](X1,X2) = X1
[a__U52](X) = X
[a__U61](X1,X2) = X1
[a__U62](X) = X
[a__U71](X) = X
[a__U81](X) = X
[a__U91](X1,X2,X3) = X1
[a__U92](X1,X2,X3) = X1
[a__U93](X1,X2,X3) = X1
[a__U94](X1,X2,X3) = X1
[a__U95](X1,X2) = X1
[a__U96](X) = X
[a__isNat](X) = 2
[a__isNatIList](X) = 2
[a__isNatIListKind](X) = 2
[a__isNatKind](X) = 2
[a__isNatList](X) = 2
[a__length](X) = X + 2
[a__take](X1,X2) = 2.X1 + 2.X2 + 2
[a__zeros] = 2
[mark](X) = X
[0] = 0
[U101](X1,X2,X3) = X1
[U102](X1,X2,X3) = X1
[U103](X1,X2,X3) = X1
[U104](X1,X2,X3) = X1
[U105](X1,X2) = X1
[U106](X) = X
[U11](X1,X2) = X1
[U111](X1,X2,X3) = X1 + X2 + 2.X3
[U112](X1,X2,X3) = X1 + X2 + 2.X3
[U113](X1,X2,X3) = X1 + X2
[U114](X1,X2) = X1 + X2
[U12](X1,X2) = X1
[U121](X1,X2) = X1 + 2.X2
[U122](X) = X
[U13](X) = X
[U131](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U132](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U133](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U134](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U135](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U136](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U21](X1,X2) = X1
[U22](X1,X2) = X1
[U23](X) = X
[U31](X1,X2) = X1
[U32](X1,X2) = X1
[U33](X) = X
[U41](X1,X2,X3) = X1
[U42](X1,X2,X3) = X1
[U43](X1,X2,X3) = X1
[U44](X1,X2,X3) = X1
[U45](X1,X2) = X1
[U46](X) = X
[U51](X1,X2) = X1
[U52](X) = X
[U61](X1,X2) = X1
[U62](X) = X
[U71](X) = X
[U81](X) = X
[U91](X1,X2,X3) = X1
[U92](X1,X2,X3) = X1
[U93](X1,X2,X3) = X1
[U94](X1,X2,X3) = X1
[U95](X1,X2) = X1
[U96](X) = X
[cons](X1,X2) = 2.X1 + X2
[isNat](X) = 2
[isNatIList](X) = 2
[isNatIListKind](X) = 2
[isNatKind](X) = 2
[isNatList](X) = 2
[length](X) = X + 2
[nil] = 2
[s](X) = X
[take](X1,X2) = 2.X1 + 2.X2 + 2
[tt] = 2
[zeros] = 2
[A__U111](X1,X2,X3) = X1 + 2.X2
[A__U112](X1,X2,X3) = X1 + 2.X2
[A__U113](X1,X2,X3) = X1 + 2.X2
[A__U114](X1,X2) = X1 + 2.X2
[A__U131](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 + 1
[A__U132](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 + 1
[A__U133](X1,X2,X3,X4) = 2.X2 + 2.X3 + 2.X4 + 1
[A__U134](X1,X2,X3,X4) = 2.X2 + X3 + 2.X4 + 1
[A__U135](X1,X2,X3,X4) = 2.X2 + X3 + 2.X4 + 1
[A__U136](X1,X2,X3,X4) = 2.X2 + X3 + 2.X4 + 1
[A__LENGTH](X) = 2.X + 2
[A__TAKE](X1,X2) = 2.X1 + 2.X2 + 2
[MARK](X) = 2.X + 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 A__U111(tt,L,N) -> A__U112(a__isNatIListKind(L),L,N)
 A__U112(tt,L,N) -> A__U113(a__isNat(N),L,N)
 A__U113(tt,L,N) -> A__U114(a__isNatKind(N),L)
 A__U114(tt,L) -> A__LENGTH(mark(L))
 A__U131(tt,IL,M,N) -> A__U132(a__isNatIListKind(IL),IL,M,N)
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
 A__LENGTH(cons(N,L)) -> A__U111(a__isNatList(L),L,N)
 A__TAKE(s(M),cons(N,IL)) -> A__U131(a__isNatIList(IL),IL,M,N)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> A__U111(mark(X1),X2,X3)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> A__U112(mark(X1),X2,X3)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> A__U113(mark(X1),X2,X3)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> A__U114(mark(X1),X2)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> A__U131(mark(X1),X2,X3,X4)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(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(X)) -> MARK(X)
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> MARK(X)
 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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U111(tt,L,N) -> A__U112(a__isNatIListKind(L),L,N)
 A__U112(tt,L,N) -> A__U113(a__isNat(N),L,N)
 A__U113(tt,L,N) -> A__U114(a__isNatKind(N),L)
 A__U114(tt,L) -> A__LENGTH(mark(L))
 A__LENGTH(cons(N,L)) -> A__U111(a__isNatList(L),L,N)
->->-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U131(tt,IL,M,N) -> A__U132(a__isNatIListKind(IL),IL,M,N)
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
 A__TAKE(s(M),cons(N,IL)) -> A__U131(a__isNatIList(IL),IL,M,N)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> A__U131(mark(X1),X2,X3,X4)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(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(X)) -> MARK(X)
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> MARK(X)
 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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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 1.3.1: 

Reduction Pair Processor:
-> Pairs:
 A__U111(tt,L,N) -> A__U112(a__isNatIListKind(L),L,N)
 A__U112(tt,L,N) -> A__U113(a__isNat(N),L,N)
 A__U113(tt,L,N) -> A__U114(a__isNatKind(N),L)
 A__U114(tt,L) -> A__LENGTH(mark(L))
 A__LENGTH(cons(N,L)) -> A__U111(a__isNatList(L),L,N)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101](X1,X2,X3) = X2
[a__U102](X1,X2,X3) = X2
[a__U103](X1,X2,X3) = X2
[a__U104](X1,X2,X3) = X2
[a__U105](X1,X2) = X1
[a__U106](X) = X
[a__U11](X1,X2) = X2
[a__U111](X1,X2,X3) = 2.X2
[a__U112](X1,X2,X3) = 2.X2
[a__U113](X1,X2,X3) = 2.X2
[a__U114](X1,X2) = 2.X2
[a__U12](X1,X2) = X2
[a__U121](X1,X2) = 2
[a__U122](X) = X
[a__U13](X) = X
[a__U131](X1,X2,X3,X4) = 2.X3
[a__U132](X1,X2,X3,X4) = 2.X3
[a__U133](X1,X2,X3,X4) = 2.X3
[a__U134](X1,X2,X3,X4) = 2.X3
[a__U135](X1,X2,X3,X4) = 2.X3
[a__U136](X1,X2,X3,X4) = 2.X3
[a__U21](X1,X2) = 2.X2
[a__U22](X1,X2) = 2.X2
[a__U23](X) = 2.X
[a__U31](X1,X2) = X1
[a__U32](X1,X2) = X1
[a__U33](X) = 2
[a__U41](X1,X2,X3) = X1
[a__U42](X1,X2,X3) = X1
[a__U43](X1,X2,X3) = X1
[a__U44](X1,X2,X3) = 2
[a__U45](X1,X2) = 2
[a__U46](X) = 2
[a__U51](X1,X2) = X1
[a__U52](X) = X
[a__U61](X1,X2) = X1
[a__U62](X) = X
[a__U71](X) = X
[a__U81](X) = X
[a__U91](X1,X2,X3) = 2.X3
[a__U92](X1,X2,X3) = 2.X3
[a__U93](X1,X2,X3) = 2.X3
[a__U94](X1,X2,X3) = 2.X3
[a__U95](X1,X2) = 2.X2
[a__U96](X) = 2.X
[a__isNat](X) = X
[a__isNatIList](X) = 2
[a__isNatIListKind](X) = 2
[a__isNatKind](X) = 2
[a__isNatList](X) = X
[a__length](X) = X
[a__take](X1,X2) = X1
[a__zeros] = 0
[mark](X) = X
[0] = 2
[U101](X1,X2,X3) = X2
[U102](X1,X2,X3) = X2
[U103](X1,X2,X3) = X2
[U104](X1,X2,X3) = X2
[U105](X1,X2) = X1
[U106](X) = X
[U11](X1,X2) = X2
[U111](X1,X2,X3) = 2.X2
[U112](X1,X2,X3) = 2.X2
[U113](X1,X2,X3) = 2.X2
[U114](X1,X2) = 2.X2
[U12](X1,X2) = X2
[U121](X1,X2) = 2
[U122](X) = X
[U13](X) = X
[U131](X1,X2,X3,X4) = 2.X3
[U132](X1,X2,X3,X4) = 2.X3
[U133](X1,X2,X3,X4) = 2.X3
[U134](X1,X2,X3,X4) = 2.X3
[U135](X1,X2,X3,X4) = 2.X3
[U136](X1,X2,X3,X4) = 2.X3
[U21](X1,X2) = 2.X2
[U22](X1,X2) = 2.X2
[U23](X) = 2.X
[U31](X1,X2) = X1
[U32](X1,X2) = X1
[U33](X) = 2
[U41](X1,X2,X3) = X1
[U42](X1,X2,X3) = X1
[U43](X1,X2,X3) = X1
[U44](X1,X2,X3) = 2
[U45](X1,X2) = 2
[U46](X) = 2
[U51](X1,X2) = X1
[U52](X) = X
[U61](X1,X2) = X1
[U62](X) = X
[U71](X) = X
[U81](X) = X
[U91](X1,X2,X3) = 2.X3
[U92](X1,X2,X3) = 2.X3
[U93](X1,X2,X3) = 2.X3
[U94](X1,X2,X3) = 2.X3
[U95](X1,X2) = 2.X2
[U96](X) = 2.X
[cons](X1,X2) = 2.X2
[isNat](X) = X
[isNatIList](X) = 2
[isNatIListKind](X) = 2
[isNatKind](X) = 2
[isNatList](X) = X
[length](X) = X
[nil] = 2
[s](X) = 2.X
[take](X1,X2) = X1
[tt] = 2
[zeros] = 0
[A__U111](X1,X2,X3) = 2.X1 + 2.X2 + 2
[A__U112](X1,X2,X3) = X1 + 2.X2 + 2
[A__U113](X1,X2,X3) = 2.X2 + 2
[A__U114](X1,X2) = X1 + 2.X2
[A__LENGTH](X) = 2.X + 2

Problem 1.3.1: 

SCC Processor:
-> Pairs:
 A__U112(tt,L,N) -> A__U113(a__isNat(N),L,N)
 A__U113(tt,L,N) -> A__U114(a__isNatKind(N),L)
 A__U114(tt,L) -> A__LENGTH(mark(L))
 A__LENGTH(cons(N,L)) -> A__U111(a__isNatList(L),L,N)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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 1.3.2: 

Reduction Pair Processor:
-> Pairs:
 A__U131(tt,IL,M,N) -> A__U132(a__isNatIListKind(IL),IL,M,N)
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
 A__TAKE(s(M),cons(N,IL)) -> A__U131(a__isNatIList(IL),IL,M,N)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> A__U131(mark(X1),X2,X3,X4)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(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(X)) -> MARK(X)
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> MARK(X)
 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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U101](X1,X2,X3) = X1
[a__U102](X1,X2,X3) = X1
[a__U103](X1,X2,X3) = X1
[a__U104](X1,X2,X3) = X1
[a__U105](X1,X2) = X1
[a__U106](X) = X
[a__U11](X1,X2) = X1
[a__U111](X1,X2,X3) = X1 + 2.X2 + 2.X3
[a__U112](X1,X2,X3) = X1 + 2.X2 + 2.X3
[a__U113](X1,X2,X3) = X1 + 2.X2 + 2.X3
[a__U114](X1,X2) = X1 + 2.X2
[a__U12](X1,X2) = X1
[a__U121](X1,X2) = X1 + 2.X2
[a__U122](X) = X
[a__U13](X) = X
[a__U131](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U132](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U133](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U134](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U135](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U136](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[a__U21](X1,X2) = X1
[a__U22](X1,X2) = X1
[a__U23](X) = X
[a__U31](X1,X2) = X1
[a__U32](X1,X2) = X1
[a__U33](X) = X
[a__U41](X1,X2,X3) = X1
[a__U42](X1,X2,X3) = X1
[a__U43](X1,X2,X3) = X1
[a__U44](X1,X2,X3) = X1
[a__U45](X1,X2) = X1
[a__U46](X) = X
[a__U51](X1,X2) = X1
[a__U52](X) = X
[a__U61](X1,X2) = X1
[a__U62](X) = X
[a__U71](X) = X
[a__U81](X) = X
[a__U91](X1,X2,X3) = X1
[a__U92](X1,X2,X3) = X1
[a__U93](X1,X2,X3) = X1
[a__U94](X1,X2,X3) = X1
[a__U95](X1,X2) = X1
[a__U96](X) = X
[a__isNat](X) = 2
[a__isNatIList](X) = 2
[a__isNatIListKind](X) = 2
[a__isNatKind](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
[U101](X1,X2,X3) = X1
[U102](X1,X2,X3) = X1
[U103](X1,X2,X3) = X1
[U104](X1,X2,X3) = X1
[U105](X1,X2) = X1
[U106](X) = X
[U11](X1,X2) = X1
[U111](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U112](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U113](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U114](X1,X2) = X1 + 2.X2
[U12](X1,X2) = X1
[U121](X1,X2) = X1 + 2.X2
[U122](X) = X
[U13](X) = X
[U131](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U132](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U133](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U134](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U135](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U136](X1,X2,X3,X4) = X1 + 2.X2 + 2.X3 + 2.X4
[U21](X1,X2) = X1
[U22](X1,X2) = X1
[U23](X) = X
[U31](X1,X2) = X1
[U32](X1,X2) = X1
[U33](X) = X
[U41](X1,X2,X3) = X1
[U42](X1,X2,X3) = X1
[U43](X1,X2,X3) = X1
[U44](X1,X2,X3) = X1
[U45](X1,X2) = X1
[U46](X) = X
[U51](X1,X2) = X1
[U52](X) = X
[U61](X1,X2) = X1
[U62](X) = X
[U71](X) = X
[U81](X) = X
[U91](X1,X2,X3) = X1
[U92](X1,X2,X3) = X1
[U93](X1,X2,X3) = X1
[U94](X1,X2,X3) = X1
[U95](X1,X2) = X1
[U96](X) = X
[cons](X1,X2) = 2.X1 + X2
[isNat](X) = 2
[isNatIList](X) = 2
[isNatIListKind](X) = 2
[isNatKind](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] = 0
[A__U131](X1,X2,X3,X4) = X1 + 2.X2 + 2.X4
[A__U132](X1,X2,X3,X4) = 2.X2 + 2.X4 + 1
[A__U133](X1,X2,X3,X4) = 2.X2 + 2.X4 + 1
[A__U134](X1,X2,X3,X4) = 2.X2 + 2.X4 + 1
[A__U135](X1,X2,X3,X4) = 2.X2 + 2.X4 + 1
[A__U136](X1,X2,X3,X4) = 2.X2 + 2.X4 + 1
[A__TAKE](X1,X2) = 2.X2 + 2
[MARK](X) = 2.X + 1

Problem 1.3.2: 

SCC Processor:
-> Pairs:
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
 A__TAKE(s(M),cons(N,IL)) -> A__U131(a__isNatIList(IL),IL,M,N)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> A__U131(mark(X1),X2,X3,X4)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(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(X)) -> MARK(X)
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> MARK(X)
 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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(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(X)) -> MARK(X)
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> MARK(X)
 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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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 1.3.2: 

Subterm Processor:
-> Pairs:
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
 MARK(U101(X1,X2,X3)) -> MARK(X1)
 MARK(U102(X1,X2,X3)) -> MARK(X1)
 MARK(U103(X1,X2,X3)) -> MARK(X1)
 MARK(U104(X1,X2,X3)) -> MARK(X1)
 MARK(U105(X1,X2)) -> MARK(X1)
 MARK(U106(X)) -> MARK(X)
 MARK(U11(X1,X2)) -> MARK(X1)
 MARK(U111(X1,X2,X3)) -> MARK(X1)
 MARK(U112(X1,X2,X3)) -> MARK(X1)
 MARK(U113(X1,X2,X3)) -> MARK(X1)
 MARK(U114(X1,X2)) -> MARK(X1)
 MARK(U12(X1,X2)) -> MARK(X1)
 MARK(U121(X1,X2)) -> MARK(X1)
 MARK(U122(X)) -> MARK(X)
 MARK(U13(X)) -> MARK(X)
 MARK(U131(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U132(X1,X2,X3,X4)) -> A__U132(mark(X1),X2,X3,X4)
 MARK(U132(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U133(X1,X2,X3,X4)) -> A__U133(mark(X1),X2,X3,X4)
 MARK(U133(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U134(X1,X2,X3,X4)) -> A__U134(mark(X1),X2,X3,X4)
 MARK(U134(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U135(X1,X2,X3,X4)) -> A__U135(mark(X1),X2,X3,X4)
 MARK(U135(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U136(X1,X2,X3,X4)) -> A__U136(mark(X1),X2,X3,X4)
 MARK(U136(X1,X2,X3,X4)) -> MARK(X1)
 MARK(U21(X1,X2)) -> MARK(X1)
 MARK(U22(X1,X2)) -> MARK(X1)
 MARK(U23(X)) -> MARK(X)
 MARK(U31(X1,X2)) -> MARK(X1)
 MARK(U32(X1,X2)) -> MARK(X1)
 MARK(U33(X)) -> MARK(X)
 MARK(U41(X1,X2,X3)) -> MARK(X1)
 MARK(U42(X1,X2,X3)) -> MARK(X1)
 MARK(U43(X1,X2,X3)) -> MARK(X1)
 MARK(U44(X1,X2,X3)) -> MARK(X1)
 MARK(U45(X1,X2)) -> MARK(X1)
 MARK(U46(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(X)) -> MARK(X)
 MARK(U81(X)) -> MARK(X)
 MARK(U91(X1,X2,X3)) -> MARK(X1)
 MARK(U92(X1,X2,X3)) -> MARK(X1)
 MARK(U93(X1,X2,X3)) -> MARK(X1)
 MARK(U94(X1,X2,X3)) -> MARK(X1)
 MARK(U95(X1,X2)) -> MARK(X1)
 MARK(U96(X)) -> MARK(X)
 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__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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__U132) = 4
 pi(A__U133) = 4
 pi(A__U134) = 4
 pi(A__U135) = 4
 pi(A__U136) = 4
 pi(MARK) = 1

Problem 1.3.2: 

SCC Processor:
-> Pairs:
 A__U132(tt,IL,M,N) -> A__U133(a__isNat(M),IL,M,N)
 A__U133(tt,IL,M,N) -> A__U134(a__isNatKind(M),IL,M,N)
 A__U134(tt,IL,M,N) -> A__U135(a__isNat(N),IL,M,N)
 A__U135(tt,IL,M,N) -> A__U136(a__isNatKind(N),IL,M,N)
 A__U136(tt,IL,M,N) -> MARK(N)
-> Rules:
 a__U101(tt,V1,V2) -> a__U102(a__isNatKind(V1),V1,V2)
 a__U101(X1,X2,X3) -> U101(X1,X2,X3)
 a__U102(tt,V1,V2) -> a__U103(a__isNatIListKind(V2),V1,V2)
 a__U102(X1,X2,X3) -> U102(X1,X2,X3)
 a__U103(tt,V1,V2) -> a__U104(a__isNatIListKind(V2),V1,V2)
 a__U103(X1,X2,X3) -> U103(X1,X2,X3)
 a__U104(tt,V1,V2) -> a__U105(a__isNat(V1),V2)
 a__U104(X1,X2,X3) -> U104(X1,X2,X3)
 a__U105(tt,V2) -> a__U106(a__isNatIList(V2))
 a__U105(X1,X2) -> U105(X1,X2)
 a__U106(tt) -> tt
 a__U106(X) -> U106(X)
 a__U11(tt,V1) -> a__U12(a__isNatIListKind(V1),V1)
 a__U11(X1,X2) -> U11(X1,X2)
 a__U111(tt,L,N) -> a__U112(a__isNatIListKind(L),L,N)
 a__U111(X1,X2,X3) -> U111(X1,X2,X3)
 a__U112(tt,L,N) -> a__U113(a__isNat(N),L,N)
 a__U112(X1,X2,X3) -> U112(X1,X2,X3)
 a__U113(tt,L,N) -> a__U114(a__isNatKind(N),L)
 a__U113(X1,X2,X3) -> U113(X1,X2,X3)
 a__U114(tt,L) -> s(a__length(mark(L)))
 a__U114(X1,X2) -> U114(X1,X2)
 a__U12(tt,V1) -> a__U13(a__isNatList(V1))
 a__U12(X1,X2) -> U12(X1,X2)
 a__U121(tt,IL) -> a__U122(a__isNatIListKind(IL))
 a__U121(X1,X2) -> U121(X1,X2)
 a__U122(tt) -> nil
 a__U122(X) -> U122(X)
 a__U13(tt) -> tt
 a__U13(X) -> U13(X)
 a__U131(tt,IL,M,N) -> a__U132(a__isNatIListKind(IL),IL,M,N)
 a__U131(X1,X2,X3,X4) -> U131(X1,X2,X3,X4)
 a__U132(tt,IL,M,N) -> a__U133(a__isNat(M),IL,M,N)
 a__U132(X1,X2,X3,X4) -> U132(X1,X2,X3,X4)
 a__U133(tt,IL,M,N) -> a__U134(a__isNatKind(M),IL,M,N)
 a__U133(X1,X2,X3,X4) -> U133(X1,X2,X3,X4)
 a__U134(tt,IL,M,N) -> a__U135(a__isNat(N),IL,M,N)
 a__U134(X1,X2,X3,X4) -> U134(X1,X2,X3,X4)
 a__U135(tt,IL,M,N) -> a__U136(a__isNatKind(N),IL,M,N)
 a__U135(X1,X2,X3,X4) -> U135(X1,X2,X3,X4)
 a__U136(tt,IL,M,N) -> cons(mark(N),take(M,IL))
 a__U136(X1,X2,X3,X4) -> U136(X1,X2,X3,X4)
 a__U21(tt,V1) -> a__U22(a__isNatKind(V1),V1)
 a__U21(X1,X2) -> U21(X1,X2)
 a__U22(tt,V1) -> a__U23(a__isNat(V1))
 a__U22(X1,X2) -> U22(X1,X2)
 a__U23(tt) -> tt
 a__U23(X) -> U23(X)
 a__U31(tt,V) -> a__U32(a__isNatIListKind(V),V)
 a__U31(X1,X2) -> U31(X1,X2)
 a__U32(tt,V) -> a__U33(a__isNatList(V))
 a__U32(X1,X2) -> U32(X1,X2)
 a__U33(tt) -> tt
 a__U33(X) -> U33(X)
 a__U41(tt,V1,V2) -> a__U42(a__isNatKind(V1),V1,V2)
 a__U41(X1,X2,X3) -> U41(X1,X2,X3)
 a__U42(tt,V1,V2) -> a__U43(a__isNatIListKind(V2),V1,V2)
 a__U42(X1,X2,X3) -> U42(X1,X2,X3)
 a__U43(tt,V1,V2) -> a__U44(a__isNatIListKind(V2),V1,V2)
 a__U43(X1,X2,X3) -> U43(X1,X2,X3)
 a__U44(tt,V1,V2) -> a__U45(a__isNat(V1),V2)
 a__U44(X1,X2,X3) -> U44(X1,X2,X3)
 a__U45(tt,V2) -> a__U46(a__isNatIList(V2))
 a__U45(X1,X2) -> U45(X1,X2)
 a__U46(tt) -> tt
 a__U46(X) -> U46(X)
 a__U51(tt,V2) -> a__U52(a__isNatIListKind(V2))
 a__U51(X1,X2) -> U51(X1,X2)
 a__U52(tt) -> tt
 a__U52(X) -> U52(X)
 a__U61(tt,V2) -> a__U62(a__isNatIListKind(V2))
 a__U61(X1,X2) -> U61(X1,X2)
 a__U62(tt) -> tt
 a__U62(X) -> U62(X)
 a__U71(tt) -> tt
 a__U71(X) -> U71(X)
 a__U81(tt) -> tt
 a__U81(X) -> U81(X)
 a__U91(tt,V1,V2) -> a__U92(a__isNatKind(V1),V1,V2)
 a__U91(X1,X2,X3) -> U91(X1,X2,X3)
 a__U92(tt,V1,V2) -> a__U93(a__isNatIListKind(V2),V1,V2)
 a__U92(X1,X2,X3) -> U92(X1,X2,X3)
 a__U93(tt,V1,V2) -> a__U94(a__isNatIListKind(V2),V1,V2)
 a__U93(X1,X2,X3) -> U93(X1,X2,X3)
 a__U94(tt,V1,V2) -> a__U95(a__isNat(V1),V2)
 a__U94(X1,X2,X3) -> U94(X1,X2,X3)
 a__U95(tt,V2) -> a__U96(a__isNatList(V2))
 a__U95(X1,X2) -> U95(X1,X2)
 a__U96(tt) -> tt
 a__U96(X) -> U96(X)
 a__isNat(0) -> tt
 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1),V1)
 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
 a__isNat(X) -> isNat(X)
 a__isNatIList(cons(V1,V2)) -> a__U41(a__isNatKind(V1),V1,V2)
 a__isNatIList(zeros) -> tt
 a__isNatIList(V) -> a__U31(a__isNatIListKind(V),V)
 a__isNatIList(X) -> isNatIList(X)
 a__isNatIListKind(cons(V1,V2)) -> a__U51(a__isNatKind(V1),V2)
 a__isNatIListKind(nil) -> tt
 a__isNatIListKind(take(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
 a__isNatIListKind(zeros) -> tt
 a__isNatIListKind(X) -> isNatIListKind(X)
 a__isNatKind(0) -> tt
 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1))
 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1))
 a__isNatKind(X) -> isNatKind(X)
 a__isNatList(cons(V1,V2)) -> a__U91(a__isNatKind(V1),V1,V2)
 a__isNatList(nil) -> tt
 a__isNatList(take(V1,V2)) -> a__U101(a__isNatKind(V1),V1,V2)
 a__isNatList(X) -> isNatList(X)
 a__length(cons(N,L)) -> a__U111(a__isNatList(L),L,N)
 a__length(nil) -> 0
 a__length(X) -> length(X)
 a__take(0,IL) -> a__U121(a__isNatIList(IL),IL)
 a__take(s(M),cons(N,IL)) -> a__U131(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(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
 mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
 mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
 mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
 mark(U105(X1,X2)) -> a__U105(mark(X1),X2)
 mark(U106(X)) -> a__U106(mark(X))
 mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
 mark(U111(X1,X2,X3)) -> a__U111(mark(X1),X2,X3)
 mark(U112(X1,X2,X3)) -> a__U112(mark(X1),X2,X3)
 mark(U113(X1,X2,X3)) -> a__U113(mark(X1),X2,X3)
 mark(U114(X1,X2)) -> a__U114(mark(X1),X2)
 mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
 mark(U121(X1,X2)) -> a__U121(mark(X1),X2)
 mark(U122(X)) -> a__U122(mark(X))
 mark(U13(X)) -> a__U13(mark(X))
 mark(U131(X1,X2,X3,X4)) -> a__U131(mark(X1),X2,X3,X4)
 mark(U132(X1,X2,X3,X4)) -> a__U132(mark(X1),X2,X3,X4)
 mark(U133(X1,X2,X3,X4)) -> a__U133(mark(X1),X2,X3,X4)
 mark(U134(X1,X2,X3,X4)) -> a__U134(mark(X1),X2,X3,X4)
 mark(U135(X1,X2,X3,X4)) -> a__U135(mark(X1),X2,X3,X4)
 mark(U136(X1,X2,X3,X4)) -> a__U136(mark(X1),X2,X3,X4)
 mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
 mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
 mark(U23(X)) -> a__U23(mark(X))
 mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
 mark(U32(X1,X2)) -> a__U32(mark(X1),X2)
 mark(U33(X)) -> a__U33(mark(X))
 mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
 mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
 mark(U43(X1,X2,X3)) -> a__U43(mark(X1),X2,X3)
 mark(U44(X1,X2,X3)) -> a__U44(mark(X1),X2,X3)
 mark(U45(X1,X2)) -> a__U45(mark(X1),X2)
 mark(U46(X)) -> a__U46(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(X)) -> a__U71(mark(X))
 mark(U81(X)) -> a__U81(mark(X))
 mark(U91(X1,X2,X3)) -> a__U91(mark(X1),X2,X3)
 mark(U92(X1,X2,X3)) -> a__U92(mark(X1),X2,X3)
 mark(U93(X1,X2,X3)) -> a__U93(mark(X1),X2,X3)
 mark(U94(X1,X2,X3)) -> a__U94(mark(X1),X2,X3)
 mark(U95(X1,X2)) -> a__U95(mark(X1),X2)
 mark(U96(X)) -> a__U96(mark(X))
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(isNat(X)) -> a__isNat(X)
 mark(isNatIList(X)) -> a__isNatIList(X)
 mark(isNatIListKind(X)) -> a__isNatIListKind(X)
 mark(isNatKind(X)) -> a__isNatKind(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.