Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MARK(U42(X1, X2, X3)) → A__U42(mark(X1), X2, X3)
MARK(U11(X1, X2)) → MARK(X1)
A__U21(tt, V1) → A__ISNATKIND(V1)
MARK(U81(X)) → A__U81(mark(X))
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U96(X)) → MARK(X)
A__U104(tt, V1, V2) → A__ISNAT(V1)
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__U136(tt, IL, M, N) → MARK(N)
A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
A__U114(tt, L) → MARK(L)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → A__U62(mark(X))
MARK(U104(X1, X2, X3)) → A__U104(mark(X1), X2, X3)
MARK(U112(X1, X2, X3)) → A__U112(mark(X1), X2, X3)
MARK(U105(X1, X2)) → A__U105(mark(X1), X2)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U44(X1, X2, X3)) → A__U44(mark(X1), X2, X3)
A__U135(tt, IL, M, N) → A__ISNATKIND(N)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U22(tt, V1) → A__U23(a__isNat(V1))
A__U103(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
MARK(U45(X1, X2)) → A__U45(mark(X1), X2)
A__U101(tt, V1, V2) → A__ISNATKIND(V1)
A__U43(tt, V1, V2) → A__ISNATILISTKIND(V2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__U12(tt, V1) → A__ISNATLIST(V1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U96(X)) → A__U96(mark(X))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
MARK(U101(X1, X2, X3)) → A__U101(mark(X1), X2, X3)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(take(X1, X2)) → MARK(X1)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U114(X1, X2)) → A__U114(mark(X1), X2)
A__U45(tt, V2) → A__ISNATILIST(V2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U51(X1, X2)) → MARK(X1)
A__U114(tt, L) → A__LENGTH(mark(L))
MARK(U103(X1, X2, X3)) → A__U103(mark(X1), X2, X3)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U93(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U113(tt, L, N) → A__ISNATKIND(N)
MARK(U94(X1, X2, X3)) → A__U94(mark(X1), X2, X3)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
MARK(U122(X)) → MARK(X)
MARK(U111(X1, X2, X3)) → A__U111(mark(X1), X2, X3)
A__U32(tt, V) → A__ISNATLIST(V)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
MARK(U71(X)) → A__U71(mark(X))
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
A__U61(tt, V2) → A__U62(a__isNatIListKind(V2))
MARK(U92(X1, X2, X3)) → MARK(X1)
A__U91(tt, V1, V2) → A__ISNATKIND(V1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → A__U33(mark(X))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATKIND(length(V1)) → A__U71(a__isNatIListKind(V1))
MARK(U23(X)) → A__U23(mark(X))
A__U102(tt, V1, V2) → A__ISNATILISTKIND(V2)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U46(X)) → A__U46(mark(X))
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U113(X1, X2, X3)) → MARK(X1)
A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U101(X1, X2, X3)) → MARK(X1)
A__U12(tt, V1) → A__U13(a__isNatList(V1))
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
MARK(U93(X1, X2, X3)) → MARK(X1)
A__U134(tt, IL, M, N) → A__ISNAT(N)
MARK(U23(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U95(X1, X2)) → MARK(X1)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U11(tt, V1) → A__ISNATILISTKIND(V1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
A__U133(tt, IL, M, N) → A__ISNATKIND(M)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
MARK(U104(X1, X2, X3)) → MARK(X1)
A__U112(tt, L, N) → A__ISNAT(N)
MARK(length(X)) → MARK(X)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U105(tt, V2) → A__ISNATILIST(V2)
MARK(U114(X1, X2)) → MARK(X1)
MARK(U71(X)) → MARK(X)
A__U121(tt, IL) → A__U122(a__isNatIListKind(IL))
MARK(U121(X1, X2)) → A__U121(mark(X1), X2)
MARK(U111(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U106(X)) → A__U106(mark(X))
A__U121(tt, IL) → A__ISNATILISTKIND(IL)
MARK(U13(X)) → A__U13(mark(X))
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
A__U31(tt, V) → A__ISNATILISTKIND(V)
MARK(U105(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNATKIND(V1)
A__U42(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
MARK(U43(X1, X2, X3)) → A__U43(mark(X1), X2, X3)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U52(X)) → MARK(X)
A__U105(tt, V2) → A__U106(a__isNatIList(V2))
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
A__U131(tt, IL, M, N) → A__ISNATILISTKIND(IL)
MARK(U33(X)) → MARK(X)
A__U111(tt, L, N) → A__ISNATILISTKIND(L)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
MARK(cons(X1, X2)) → MARK(X1)
A__U92(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
MARK(U112(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U45(tt, V2) → A__U46(a__isNatIList(V2))
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
A__U32(tt, V) → A__U33(a__isNatList(V))
MARK(U113(X1, X2, X3)) → A__U113(mark(X1), X2, X3)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(zeros) → A__ZEROS
MARK(U31(X1, X2)) → MARK(X1)
A__U95(tt, V2) → A__U96(a__isNatList(V2))
MARK(U81(X)) → MARK(X)
A__U51(tt, V2) → A__U52(a__isNatIListKind(V2))
A__TAKE(0, IL) → A__U121(a__isNatIList(IL), IL)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U43(X1, X2, X3)) → MARK(X1)
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
MARK(U52(X)) → A__U52(mark(X))
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U122(X)) → A__U122(mark(X))
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U95(X1, X2)) → A__U95(mark(X1), X2)
A__U22(tt, V1) → A__ISNAT(V1)
A__ISNATKIND(s(V1)) → A__U81(a__isNatKind(V1))
A__U132(tt, IL, M, N) → A__ISNAT(M)
MARK(U102(X1, X2, X3)) → A__U102(mark(X1), X2, X3)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(U42(X1, X2, X3)) → A__U42(mark(X1), X2, X3)
MARK(U11(X1, X2)) → MARK(X1)
A__U21(tt, V1) → A__ISNATKIND(V1)
MARK(U81(X)) → A__U81(mark(X))
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U96(X)) → MARK(X)
A__U104(tt, V1, V2) → A__ISNAT(V1)
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__U136(tt, IL, M, N) → MARK(N)
A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
A__U114(tt, L) → MARK(L)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → A__U62(mark(X))
MARK(U104(X1, X2, X3)) → A__U104(mark(X1), X2, X3)
MARK(U112(X1, X2, X3)) → A__U112(mark(X1), X2, X3)
MARK(U105(X1, X2)) → A__U105(mark(X1), X2)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U44(X1, X2, X3)) → A__U44(mark(X1), X2, X3)
A__U135(tt, IL, M, N) → A__ISNATKIND(N)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U13(X)) → MARK(X)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U22(tt, V1) → A__U23(a__isNat(V1))
A__U103(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
MARK(U45(X1, X2)) → A__U45(mark(X1), X2)
A__U101(tt, V1, V2) → A__ISNATKIND(V1)
A__U43(tt, V1, V2) → A__ISNATILISTKIND(V2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__U12(tt, V1) → A__ISNATLIST(V1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U96(X)) → A__U96(mark(X))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
MARK(U101(X1, X2, X3)) → A__U101(mark(X1), X2, X3)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(take(X1, X2)) → MARK(X1)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U114(X1, X2)) → A__U114(mark(X1), X2)
A__U45(tt, V2) → A__ISNATILIST(V2)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U51(X1, X2)) → MARK(X1)
A__U114(tt, L) → A__LENGTH(mark(L))
MARK(U103(X1, X2, X3)) → A__U103(mark(X1), X2, X3)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U93(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U113(tt, L, N) → A__ISNATKIND(N)
MARK(U94(X1, X2, X3)) → A__U94(mark(X1), X2, X3)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
MARK(U122(X)) → MARK(X)
MARK(U111(X1, X2, X3)) → A__U111(mark(X1), X2, X3)
A__U32(tt, V) → A__ISNATLIST(V)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
MARK(U71(X)) → A__U71(mark(X))
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
A__U61(tt, V2) → A__U62(a__isNatIListKind(V2))
MARK(U92(X1, X2, X3)) → MARK(X1)
A__U91(tt, V1, V2) → A__ISNATKIND(V1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → A__U33(mark(X))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATKIND(length(V1)) → A__U71(a__isNatIListKind(V1))
MARK(U23(X)) → A__U23(mark(X))
A__U102(tt, V1, V2) → A__ISNATILISTKIND(V2)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U46(X)) → A__U46(mark(X))
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U113(X1, X2, X3)) → MARK(X1)
A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
MARK(U101(X1, X2, X3)) → MARK(X1)
A__U12(tt, V1) → A__U13(a__isNatList(V1))
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
MARK(U93(X1, X2, X3)) → MARK(X1)
A__U134(tt, IL, M, N) → A__ISNAT(N)
MARK(U23(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U95(X1, X2)) → MARK(X1)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U11(tt, V1) → A__ISNATILISTKIND(V1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
A__U133(tt, IL, M, N) → A__ISNATKIND(M)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
MARK(U104(X1, X2, X3)) → MARK(X1)
A__U112(tt, L, N) → A__ISNAT(N)
MARK(length(X)) → MARK(X)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U105(tt, V2) → A__ISNATILIST(V2)
MARK(U114(X1, X2)) → MARK(X1)
MARK(U71(X)) → MARK(X)
A__U121(tt, IL) → A__U122(a__isNatIListKind(IL))
MARK(U121(X1, X2)) → A__U121(mark(X1), X2)
MARK(U111(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U106(X)) → A__U106(mark(X))
A__U121(tt, IL) → A__ISNATILISTKIND(IL)
MARK(U13(X)) → A__U13(mark(X))
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
A__U31(tt, V) → A__ISNATILISTKIND(V)
MARK(U105(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNATKIND(V1)
A__U42(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
MARK(U43(X1, X2, X3)) → A__U43(mark(X1), X2, X3)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U52(X)) → MARK(X)
A__U105(tt, V2) → A__U106(a__isNatIList(V2))
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
A__U131(tt, IL, M, N) → A__ISNATILISTKIND(IL)
MARK(U33(X)) → MARK(X)
A__U111(tt, L, N) → A__ISNATILISTKIND(L)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
MARK(cons(X1, X2)) → MARK(X1)
A__U92(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
MARK(U112(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U45(tt, V2) → A__U46(a__isNatIList(V2))
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
A__U32(tt, V) → A__U33(a__isNatList(V))
MARK(U113(X1, X2, X3)) → A__U113(mark(X1), X2, X3)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(zeros) → A__ZEROS
MARK(U31(X1, X2)) → MARK(X1)
A__U95(tt, V2) → A__U96(a__isNatList(V2))
MARK(U81(X)) → MARK(X)
A__U51(tt, V2) → A__U52(a__isNatIListKind(V2))
A__TAKE(0, IL) → A__U121(a__isNatIList(IL), IL)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U43(X1, X2, X3)) → MARK(X1)
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
MARK(U52(X)) → A__U52(mark(X))
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U122(X)) → A__U122(mark(X))
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U95(X1, X2)) → A__U95(mark(X1), X2)
A__U22(tt, V1) → A__ISNAT(V1)
A__ISNATKIND(s(V1)) → A__U81(a__isNatKind(V1))
A__U132(tt, IL, M, N) → A__ISNAT(M)
MARK(U102(X1, X2, X3)) → A__U102(mark(X1), X2, X3)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 83 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__U51(tt, V2) → A__ISNATILISTKIND(V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
The remaining pairs can at least be oriented weakly.

A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
Used ordering: Polynomial interpretation [25,35]:

POL(A__ISNATILISTKIND(x1)) = (4)x_1   
POL(a__U52(x1)) = 1 + (2)x_1   
POL(A__ISNATKIND(x1)) = (4)x_1   
POL(U81(x1)) = 3 + (3)x_1   
POL(take(x1, x2)) = (2)x_1 + (4)x_2   
POL(isNatIListKind(x1)) = 0   
POL(A__U61(x1, x2)) = (4)x_2   
POL(U51(x1, x2)) = 4 + (2)x_1 + (4)x_2   
POL(tt) = 1   
POL(zeros) = 1   
POL(U52(x1)) = 2 + x_1   
POL(s(x1)) = 1 + x_1   
POL(U71(x1)) = 3 + (4)x_1   
POL(nil) = 2   
POL(a__U71(x1)) = 1 + (2)x_1   
POL(a__U51(x1, x2)) = 1 + (3)x_1 + (2)x_2   
POL(a__U62(x1)) = 3 + x_1   
POL(a__U61(x1, x2)) = 2 + (4)x_1 + x_2   
POL(isNatKind(x1)) = 4 + (3)x_1   
POL(0) = 4   
POL(a__U81(x1)) = 1   
POL(U62(x1)) = 4 + (3)x_1   
POL(cons(x1, x2)) = 3 + x_1 + (4)x_2   
POL(a__isNatKind(x1)) = 1 + (4)x_1   
POL(U61(x1, x2)) = 1 + (2)x_1 + (4)x_2   
POL(A__U51(x1, x2)) = 2 + (4)x_2   
POL(a__isNatIListKind(x1)) = (2)x_1   
POL(length(x1)) = 1 + (2)x_1   
The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(A__ISNATILISTKIND(x1)) = (2)x_1   
POL(a__U52(x1)) = 0   
POL(U81(x1)) = 3 + (3)x_1   
POL(take(x1, x2)) = 3 + (2)x_1 + (2)x_2   
POL(U51(x1, x2)) = 3 + (3)x_1 + x_2   
POL(isNatIListKind(x1)) = 1 + (3)x_1   
POL(A__U61(x1, x2)) = 2 + (2)x_1 + (3)x_2   
POL(tt) = 0   
POL(zeros) = 3   
POL(U52(x1)) = 0   
POL(s(x1)) = 4 + (4)x_1   
POL(U71(x1)) = 3   
POL(nil) = 3   
POL(a__U51(x1, x2)) = 4 + (4)x_1 + (4)x_2   
POL(a__U71(x1)) = 4   
POL(a__U62(x1)) = 4   
POL(a__U61(x1, x2)) = 4 + (4)x_1 + (4)x_2   
POL(isNatKind(x1)) = x_1   
POL(0) = 0   
POL(a__U81(x1)) = 4 + (4)x_1   
POL(U62(x1)) = 1   
POL(cons(x1, x2)) = 3 + (4)x_1 + (4)x_2   
POL(a__isNatKind(x1)) = 1 + (2)x_1   
POL(U61(x1, x2)) = 3 + x_1 + (3)x_2   
POL(a__isNatIListKind(x1)) = 1 + (4)x_1   
POL(length(x1)) = 4 + (4)x_1   
The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

a__U81(tt) → tt
a__U71(tt) → tt
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatKind(X) → isNatKind(X)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U105(tt, V2) → A__ISNATILIST(V2)
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U104(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U22(tt, V1) → A__ISNAT(V1)
A__U32(tt, V) → A__ISNATLIST(V)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U104(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)
The remaining pairs can at least be oriented weakly.

A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U105(tt, V2) → A__ISNATILIST(V2)
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U22(tt, V1) → A__ISNAT(V1)
A__U32(tt, V) → A__ISNATLIST(V)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U95(tt, V2) → A__ISNATLIST(V2)
Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x1)) = 3   
POL(A__U45(x1, x2)) = 3 + (2)x_2   
POL(A__ISNAT(x1)) = 2 + (2)x_1   
POL(U41(x1, x2, x3)) = 4 + (3)x_1 + (2)x_2 + (4)x_3   
POL(U81(x1)) = 3 + (3)x_1   
POL(U32(x1, x2)) = 3 + (2)x_1 + (2)x_2   
POL(a__isNatIList(x1)) = 4 + (2)x_1   
POL(U103(x1, x2, x3)) = 2 + (3)x_1 + x_2 + (3)x_3   
POL(take(x1, x2)) = 4 + (3)x_1 + (2)x_2   
POL(a__U105(x1, x2)) = 3 + (2)x_1 + (2)x_2   
POL(U51(x1, x2)) = 2 + (3)x_1 + (3)x_2   
POL(U101(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U22(x1, x2)) = 2 + (3)x_1 + (3)x_2   
POL(U106(x1)) = 1 + (3)x_1   
POL(tt) = 0   
POL(a__U46(x1)) = 3 + (4)x_1   
POL(A__U44(x1, x2, x3)) = 3 + (2)x_2 + (2)x_3   
POL(a__U33(x1)) = 1 + x_1   
POL(a__U21(x1, x2)) = 4 + (3)x_1 + (2)x_2   
POL(A__U32(x1, x2)) = 2 + (2)x_2   
POL(A__U91(x1, x2, x3)) = 2 + (4)x_2 + (2)x_3   
POL(nil) = 4   
POL(A__U11(x1, x2)) = 2 + (2)x_2   
POL(a__U103(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (4)x_3   
POL(U22(x1, x2)) = 1 + (4)x_1 + (3)x_2   
POL(isNatKind(x1)) = 1   
POL(A__U104(x1, x2, x3)) = 3 + (2)x_2 + (2)x_3   
POL(a__U104(x1, x2, x3)) = 1 + (2)x_1 + (2)x_2 + (3)x_3   
POL(a__U31(x1, x2)) = 4 + (3)x_1 + (3)x_2   
POL(A__U95(x1, x2)) = 2 + (2)x_2   
POL(A__U92(x1, x2, x3)) = 2 + (2)x_2 + (2)x_3   
POL(length(x1)) = x_1   
POL(U95(x1, x2)) = 3 + (3)x_1 + (2)x_2   
POL(a__U91(x1, x2, x3)) = 3 + (3)x_1 + x_2 + (3)x_3   
POL(U43(x1, x2, x3)) = 1 + (3)x_1 + (3)x_2 + x_3   
POL(A__U93(x1, x2, x3)) = 2 + (2)x_2 + (2)x_3   
POL(U104(x1, x2, x3)) = 2 + (2)x_1 + (3)x_2 + (3)x_3   
POL(a__U101(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(isNatList(x1)) = 1 + (4)x_1   
POL(U52(x1)) = 3 + (3)x_1   
POL(isNatIList(x1)) = 1 + (2)x_1   
POL(a__isNat(x1)) = x_1   
POL(U71(x1)) = 3 + x_1   
POL(a__U51(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(a__U71(x1)) = 2   
POL(a__U43(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(A__U12(x1, x2)) = 2 + (2)x_2   
POL(a__U61(x1, x2)) = 2 + (3)x_1 + (4)x_2   
POL(A__U22(x1, x2)) = 2 + (3)x_2   
POL(A__U102(x1, x2, x3)) = 3 + (4)x_2 + (2)x_3   
POL(a__U81(x1)) = 2 + (4)x_1   
POL(U91(x1, x2, x3)) = 4 + x_1 + (3)x_2 + (3)x_3   
POL(a__U41(x1, x2, x3)) = 2 + x_1 + (3)x_2 + (3)x_3   
POL(A__U21(x1, x2)) = 4 + (3)x_2   
POL(A__U94(x1, x2, x3)) = 2 + (2)x_2 + (2)x_3   
POL(a__isNatKind(x1)) = 0   
POL(U61(x1, x2)) = 1 + x_1 + (3)x_2   
POL(U46(x1)) = 3 + (3)x_1   
POL(U13(x1)) = 3 + (4)x_1   
POL(U92(x1, x2, x3)) = 2 + (3)x_1 + x_2 + (2)x_3   
POL(a__U92(x1, x2, x3)) = 3 + (3)x_1 + (2)x_2 + x_3   
POL(A__U41(x1, x2, x3)) = 3 + (4)x_2 + (2)x_3   
POL(A__U103(x1, x2, x3)) = 3 + (2)x_2 + (2)x_3   
POL(a__U32(x1, x2)) = 3 + x_1 + (3)x_2   
POL(A__U31(x1, x2)) = 3 + (2)x_2   
POL(a__U11(x1, x2)) = 4 + (3)x_1 + (2)x_2   
POL(U12(x1, x2)) = 3 + (2)x_1 + (3)x_2   
POL(A__U43(x1, x2, x3)) = 3 + (2)x_2 + (2)x_3   
POL(a__U95(x1, x2)) = 3 + (4)x_1 + (2)x_2   
POL(A__ISNATLIST(x1)) = 2 + (2)x_1   
POL(U23(x1)) = 3 + (3)x_1   
POL(a__U102(x1, x2, x3)) = 1 + (3)x_1 + x_3   
POL(U93(x1, x2, x3)) = 4 + (3)x_1 + x_2 + (3)x_3   
POL(a__U106(x1)) = 3 + (2)x_1   
POL(a__U13(x1)) = 4   
POL(U105(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(a__U62(x1)) = 2 + x_1   
POL(U31(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(U102(x1, x2, x3)) = 2 + (2)x_1 + (4)x_2 + (3)x_3   
POL(U62(x1)) = 3 + (3)x_1   
POL(A__ISNATILIST(x1)) = 3 + (2)x_1   
POL(U44(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U45(x1, x2)) = 1 + (2)x_1 + x_2   
POL(U33(x1)) = 3 + (2)x_1   
POL(U94(x1, x2, x3)) = 3 + x_1 + x_2 + (3)x_3   
POL(A__U42(x1, x2, x3)) = 3 + (3)x_2 + (2)x_3   
POL(U21(x1, x2)) = 2 + (3)x_1 + (3)x_2   
POL(isNatIListKind(x1)) = 2 + (4)x_1   
POL(a__U23(x1)) = 3 + (3)x_1   
POL(U42(x1, x2, x3)) = 4 + x_1 + (2)x_2 + (3)x_3   
POL(zeros) = 1   
POL(A__U105(x1, x2)) = 3 + (2)x_2   
POL(U45(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(s(x1)) = 4 + (3)x_1   
POL(a__U96(x1)) = 3   
POL(isNat(x1)) = 3 + x_1   
POL(a__isNatList(x1)) = 3 + (4)x_1   
POL(a__U94(x1, x2, x3)) = 3 + (4)x_1 + (3)x_2 + (4)x_3   
POL(a__U93(x1, x2, x3)) = 3 + (2)x_1 + (3)x_2 + x_3   
POL(a__U44(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(U11(x1, x2)) = 3 + (3)x_1 + (2)x_2   
POL(A__U101(x1, x2, x3)) = 4 + (4)x_2 + (2)x_3   
POL(0) = 3   
POL(cons(x1, x2)) = (4)x_1 + x_2   
POL(a__U42(x1, x2, x3)) = 3 + (2)x_1 + x_2 + (3)x_3   
POL(a__isNatIListKind(x1)) = 0   
POL(a__U12(x1, x2)) = 1 + (2)x_2   
POL(U96(x1)) = 3 + (3)x_1   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U22(tt, V1) → A__ISNAT(V1)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U32(tt, V) → A__ISNATLIST(V)
A__U105(tt, V2) → A__ISNATILIST(V2)
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                      ↳ QDPOrderProof
                    ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
The remaining pairs can at least be oriented weakly.

A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U94(tt, V1, V2) → A__U95(a__isNat(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)
Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x1)) = 4   
POL(A__ISNAT(x1)) = 2 + x_1   
POL(U81(x1)) = 4 + x_1   
POL(U41(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U11(x1, x2)) = 2 + (3)x_2   
POL(U32(x1, x2)) = 2 + (2)x_1 + (3)x_2   
POL(a__isNatIList(x1)) = 2 + (2)x_1   
POL(U12(x1, x2)) = 1 + (2)x_1 + (3)x_2   
POL(a__U95(x1, x2)) = 4 + (2)x_1 + (3)x_2   
POL(take(x1, x2)) = 4   
POL(U103(x1, x2, x3)) = 1 + (2)x_1 + (2)x_3   
POL(a__U105(x1, x2)) = 2 + (3)x_1 + (2)x_2   
POL(A__ISNATLIST(x1)) = (2)x_1   
POL(U51(x1, x2)) = 3   
POL(U101(x1, x2, x3)) = 3 + (3)x_1 + (4)x_2   
POL(a__U22(x1, x2)) = 1 + (2)x_1 + (4)x_2   
POL(U23(x1)) = 3 + x_1   
POL(U106(x1)) = 2 + (2)x_1   
POL(a__U102(x1, x2, x3)) = 2 + (2)x_1 + (3)x_2 + (4)x_3   
POL(tt) = 4   
POL(a__U46(x1)) = 3 + (3)x_1   
POL(a__U33(x1)) = 3 + x_1   
POL(U93(x1, x2, x3)) = 3 + x_1 + (3)x_2 + (3)x_3   
POL(a__U21(x1, x2)) = 3 + (4)x_1 + (3)x_2   
POL(U105(x1, x2)) = 3 + (2)x_1 + x_2   
POL(a__U13(x1)) = 4 + (2)x_1   
POL(a__U106(x1)) = (4)x_1   
POL(A__U91(x1, x2, x3)) = x_1 + (4)x_2 + (4)x_3   
POL(nil) = 0   
POL(a__U62(x1)) = x_1   
POL(U31(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(A__U11(x1, x2)) = 4 + (2)x_2   
POL(a__U103(x1, x2, x3)) = 1 + (3)x_1 + (3)x_2 + (2)x_3   
POL(U22(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(U102(x1, x2, x3)) = 1 + x_1 + (2)x_2 + (3)x_3   
POL(isNatKind(x1)) = (3)x_1   
POL(a__U104(x1, x2, x3)) = 1 + x_1 + (3)x_2 + (3)x_3   
POL(U62(x1)) = 0   
POL(A__U95(x1, x2)) = 2 + (2)x_2   
POL(a__U31(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(U44(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U45(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(U33(x1)) = 2 + (4)x_1   
POL(A__U92(x1, x2, x3)) = 2 + (4)x_2 + (4)x_3   
POL(length(x1)) = 4 + (2)x_1   
POL(U95(x1, x2)) = 3 + (3)x_1 + x_2   
POL(U94(x1, x2, x3)) = 3 + x_1 + (3)x_2 + (3)x_3   
POL(a__U91(x1, x2, x3)) = 3 + x_1 + (3)x_2 + (3)x_3   
POL(U43(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (2)x_3   
POL(A__U93(x1, x2, x3)) = 2 + (2)x_2 + (4)x_3   
POL(isNatIListKind(x1)) = 3   
POL(U21(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(a__U23(x1)) = 4 + x_1   
POL(U104(x1, x2, x3)) = 1 + (2)x_1 + x_2 + (2)x_3   
POL(a__U101(x1, x2, x3)) = 3 + (3)x_1 + (4)x_2 + (3)x_3   
POL(U42(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(isNatList(x1)) = 2 + (2)x_1   
POL(zeros) = 2   
POL(U52(x1)) = 2   
POL(isNatIList(x1)) = 1 + x_1   
POL(a__isNat(x1)) = x_1   
POL(s(x1)) = 3 + (4)x_1   
POL(U45(x1, x2)) = 3 + (4)x_1 + x_2   
POL(a__U96(x1)) = 1 + (3)x_1   
POL(isNat(x1)) = 3 + (3)x_1   
POL(a__isNatList(x1)) = (3)x_1   
POL(U71(x1)) = (3)x_1   
POL(a__U71(x1)) = (4)x_1   
POL(a__U51(x1, x2)) = 4   
POL(a__U94(x1, x2, x3)) = 2 + x_1 + (2)x_2 + (2)x_3   
POL(a__U43(x1, x2, x3)) = 4 + (2)x_1 + x_2 + (2)x_3   
POL(A__U12(x1, x2)) = 3 + (2)x_2   
POL(a__U93(x1, x2, x3)) = 2 + x_1 + (4)x_2 + x_3   
POL(a__U61(x1, x2)) = 4   
POL(a__U44(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(U11(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(0) = 4   
POL(a__U81(x1)) = 4 + x_1   
POL(cons(x1, x2)) = 1 + (4)x_1 + (2)x_2   
POL(U91(x1, x2, x3)) = 3 + (4)x_1 + (3)x_2 + (3)x_3   
POL(a__U41(x1, x2, x3)) = 3 + (2)x_1 + (3)x_2 + (3)x_3   
POL(A__U94(x1, x2, x3)) = 2 + x_2 + (3)x_3   
POL(a__U42(x1, x2, x3)) = 3 + (2)x_1 + (3)x_2 + (2)x_3   
POL(a__isNatKind(x1)) = (4)x_1   
POL(U61(x1, x2)) = 3   
POL(U46(x1)) = 1 + x_1   
POL(U13(x1)) = 1 + (2)x_1   
POL(U92(x1, x2, x3)) = 1 + (2)x_1 + (2)x_2 + (4)x_3   
POL(a__U92(x1, x2, x3)) = 4 + (2)x_1 + (3)x_2 + (2)x_3   
POL(a__isNatIListKind(x1)) = 4   
POL(a__U12(x1, x2)) = 1 + (2)x_1 + (3)x_2   
POL(a__U32(x1, x2)) = 3 + (3)x_1 + (2)x_2   
POL(U96(x1)) = 1 + (2)x_1   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
a__U81(tt) → tt
a__U71(tt) → tt
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(nil) → tt
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                      ↳ QDPOrderProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                    ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 4 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                      ↳ QDPOrderProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
The remaining pairs can at least be oriented weakly.

A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x1)) = 2 + (2)x_1   
POL(A__U45(x1, x2)) = (2)x_2   
POL(U81(x1)) = 2 + (2)x_1   
POL(U41(x1, x2, x3)) = 3 + (4)x_1 + (2)x_2 + (3)x_3   
POL(a__U11(x1, x2)) = 4 + x_1 + (4)x_2   
POL(U32(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(a__isNatIList(x1)) = 3 + (4)x_1   
POL(U12(x1, x2)) = 3 + x_1 + (3)x_2   
POL(a__U95(x1, x2)) = 3 + (2)x_1 + x_2   
POL(A__U43(x1, x2, x3)) = (2)x_3   
POL(take(x1, x2)) = 2 + (3)x_1 + (3)x_2   
POL(U103(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (4)x_3   
POL(a__U105(x1, x2)) = 2 + (2)x_1 + (4)x_2   
POL(U51(x1, x2)) = (4)x_1 + (2)x_2   
POL(U101(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U22(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(U23(x1)) = 3 + x_1   
POL(U106(x1)) = 3 + (3)x_1   
POL(a__U102(x1, x2, x3)) = 3 + (3)x_2 + (3)x_3   
POL(tt) = 0   
POL(a__U46(x1)) = 4   
POL(a__U33(x1)) = 3 + (4)x_1   
POL(A__U44(x1, x2, x3)) = (2)x_3   
POL(U93(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U21(x1, x2)) = 2 + (3)x_1 + (3)x_2   
POL(a__U106(x1)) = 3   
POL(a__U13(x1)) = 1 + (2)x_1   
POL(U105(x1, x2)) = 3 + x_1 + (4)x_2   
POL(nil) = 0   
POL(a__U62(x1)) = 3 + (3)x_1   
POL(U31(x1, x2)) = 3 + (4)x_1 + (4)x_2   
POL(a__U103(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(U22(x1, x2)) = 2 + (3)x_1 + (3)x_2   
POL(U102(x1, x2, x3)) = 3 + (2)x_1 + (3)x_2 + (3)x_3   
POL(isNatKind(x1)) = 3 + (2)x_1   
POL(a__U104(x1, x2, x3)) = 2 + (2)x_2 + (3)x_3   
POL(U62(x1)) = 4 + (3)x_1   
POL(A__ISNATILIST(x1)) = (2)x_1   
POL(a__U31(x1, x2)) = 3 + (3)x_2   
POL(U44(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U45(x1, x2)) = 3 + (4)x_1 + (2)x_2   
POL(U33(x1)) = 3 + (3)x_1   
POL(length(x1)) = 4 + (3)x_1   
POL(U95(x1, x2)) = 2 + x_1 + x_2   
POL(U94(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(A__U42(x1, x2, x3)) = 4 + (2)x_3   
POL(a__U91(x1, x2, x3)) = 3 + x_2 + (3)x_3   
POL(U43(x1, x2, x3)) = 1 + (4)x_1 + (2)x_2 + (2)x_3   
POL(isNatIListKind(x1)) = 1   
POL(U21(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(a__U23(x1)) = 4 + (2)x_1   
POL(U104(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U101(x1, x2, x3)) = 4 + (3)x_1 + (3)x_2 + (3)x_3   
POL(U42(x1, x2, x3)) = 1 + (3)x_1 + (3)x_2 + (3)x_3   
POL(isNatList(x1)) = 3 + (3)x_1   
POL(zeros) = 0   
POL(U52(x1)) = 3 + (3)x_1   
POL(isNatIList(x1)) = (3)x_1   
POL(a__isNat(x1)) = 0   
POL(s(x1)) = 3 + (3)x_1   
POL(U45(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(a__U96(x1)) = 2 + (3)x_1   
POL(isNat(x1)) = 3 + (3)x_1   
POL(a__isNatList(x1)) = 4 + (4)x_1   
POL(U71(x1)) = 2 + x_1   
POL(a__U71(x1)) = 3 + (3)x_1   
POL(a__U51(x1, x2)) = 1 + (4)x_1 + (4)x_2   
POL(a__U94(x1, x2, x3)) = 3 + (3)x_2 + (2)x_3   
POL(a__U43(x1, x2, x3)) = 2 + (3)x_1 + (3)x_2 + x_3   
POL(a__U93(x1, x2, x3)) = 3 + x_1 + (3)x_2 + (3)x_3   
POL(a__U61(x1, x2)) = 2 + (3)x_1 + x_2   
POL(a__U44(x1, x2, x3)) = 3 + (3)x_1 + (2)x_2 + (2)x_3   
POL(U11(x1, x2)) = 1 + (2)x_1 + (3)x_2   
POL(0) = 3   
POL(a__U81(x1)) = 1   
POL(cons(x1, x2)) = 2 + (4)x_1 + x_2   
POL(U91(x1, x2, x3)) = 2 + (3)x_1 + (2)x_2 + (3)x_3   
POL(a__U41(x1, x2, x3)) = 4 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U42(x1, x2, x3)) = 1 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__isNatKind(x1)) = 1   
POL(U61(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(U46(x1)) = 2 + (3)x_1   
POL(U13(x1)) = 2 + (3)x_1   
POL(U92(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U92(x1, x2, x3)) = 3 + (3)x_2 + (3)x_3   
POL(A__U41(x1, x2, x3)) = 4 + (3)x_2 + (2)x_3   
POL(a__isNatIListKind(x1)) = 1   
POL(a__U12(x1, x2)) = 3 + (3)x_2   
POL(a__U32(x1, x2)) = 3 + (2)x_1 + (3)x_2   
POL(U96(x1)) = 2 + (4)x_1   
The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                      ↳ QDPOrderProof
                        ↳ QDP
                          ↳ DependencyGraphProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(cons(X1, X2)) → MARK(X1)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U112(X1, X2, X3)) → MARK(X1)
MARK(length(X)) → MARK(X)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X1)
MARK(U71(X)) → MARK(X)
MARK(U114(X1, X2)) → MARK(X1)
MARK(U114(X1, X2)) → A__U114(mark(X1), X2)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U51(X1, X2)) → MARK(X1)
A__U114(tt, L) → A__LENGTH(mark(L))
MARK(U111(X1, X2, X3)) → MARK(X1)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U96(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
A__U136(tt, IL, M, N) → MARK(N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
MARK(U122(X)) → MARK(X)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U113(X1, X2, X3)) → A__U113(mark(X1), X2, X3)
A__U114(tt, L) → MARK(L)
MARK(U111(X1, X2, X3)) → A__U111(mark(X1), X2, X3)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U112(X1, X2, X3)) → A__U112(mark(X1), X2, X3)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U81(X)) → MARK(X)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(U113(X1, X2, X3)) → MARK(X1)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U112(X1, X2, X3)) → MARK(X1)
MARK(length(X)) → MARK(X)
MARK(U114(X1, X2)) → MARK(X1)
MARK(U111(X1, X2, X3)) → MARK(X1)
A__U114(tt, L) → MARK(L)
MARK(U113(X1, X2, X3)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(cons(X1, X2)) → MARK(X1)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X1)
MARK(U71(X)) → MARK(X)
MARK(U114(X1, X2)) → A__U114(mark(X1), X2)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U51(X1, X2)) → MARK(X1)
A__U114(tt, L) → A__LENGTH(mark(L))
MARK(U121(X1, X2)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U96(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
A__U136(tt, IL, M, N) → MARK(N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
MARK(U122(X)) → MARK(X)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U113(X1, X2, X3)) → A__U113(mark(X1), X2, X3)
MARK(U111(X1, X2, X3)) → A__U111(mark(X1), X2, X3)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U112(X1, X2, X3)) → A__U112(mark(X1), X2, X3)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U81(X)) → MARK(X)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)
Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x1)) = x_1   
POL(a__zeros) = 0   
POL(U81(x1)) = (4)x_1   
POL(U41(x1, x2, x3)) = (2)x_1   
POL(U32(x1, x2)) = (4)x_1   
POL(mark(x1)) = x_1   
POL(a__isNatIList(x1)) = 0   
POL(U103(x1, x2, x3)) = (4)x_1   
POL(take(x1, x2)) = x_1 + (2)x_2   
POL(a__U105(x1, x2)) = (4)x_1   
POL(A__U132(x1, x2, x3, x4)) = x_1 + x_4   
POL(U51(x1, x2)) = x_1   
POL(a__U22(x1, x2)) = (2)x_1   
POL(U101(x1, x2, x3)) = (2)x_1   
POL(U106(x1)) = (4)x_1   
POL(a__U46(x1)) = (2)x_1   
POL(tt) = 0   
POL(a__U33(x1)) = x_1   
POL(a__U21(x1, x2)) = x_1   
POL(U113(x1, x2, x3)) = 1 + x_1 + x_2 + x_3   
POL(nil) = 0   
POL(a__U134(x1, x2, x3, x4)) = x_1 + (2)x_2 + x_3 + x_4   
POL(a__U103(x1, x2, x3)) = (4)x_1   
POL(U22(x1, x2)) = (2)x_1   
POL(isNatKind(x1)) = 0   
POL(a__U104(x1, x2, x3)) = x_1   
POL(A__TAKE(x1, x2)) = x_1 + (2)x_2   
POL(A__U111(x1, x2, x3)) = 1 + x_2 + x_3   
POL(a__U31(x1, x2)) = x_1   
POL(a__U113(x1, x2, x3)) = 1 + x_1 + x_2 + x_3   
POL(A__U136(x1, x2, x3, x4)) = x_4   
POL(length(x1)) = 1 + x_1   
POL(U95(x1, x2)) = x_1   
POL(U112(x1, x2, x3)) = 1 + (4)x_1 + x_2 + x_3   
POL(a__U91(x1, x2, x3)) = (2)x_1   
POL(U43(x1, x2, x3)) = (4)x_1   
POL(U122(x1)) = (2)x_1   
POL(U104(x1, x2, x3)) = x_1   
POL(a__U101(x1, x2, x3)) = (2)x_1   
POL(a__U121(x1, x2)) = (4)x_1 + x_2   
POL(isNatList(x1)) = 0   
POL(U52(x1)) = x_1   
POL(isNatIList(x1)) = 0   
POL(a__isNat(x1)) = 0   
POL(U71(x1)) = x_1   
POL(a__U51(x1, x2)) = x_1   
POL(a__U71(x1)) = x_1   
POL(a__U43(x1, x2, x3)) = (4)x_1   
POL(a__U61(x1, x2)) = x_1   
POL(a__U114(x1, x2)) = 1 + x_1 + x_2   
POL(a__U122(x1)) = (2)x_1   
POL(a__take(x1, x2)) = x_1 + (2)x_2   
POL(a__U81(x1)) = (4)x_1   
POL(a__U41(x1, x2, x3)) = (2)x_1   
POL(U91(x1, x2, x3)) = (2)x_1   
POL(A__U134(x1, x2, x3, x4)) = x_4   
POL(MARK(x1)) = x_1   
POL(a__isNatKind(x1)) = 0   
POL(U61(x1, x2)) = x_1   
POL(U46(x1)) = (2)x_1   
POL(U13(x1)) = (4)x_1   
POL(U92(x1, x2, x3)) = (4)x_1   
POL(a__U92(x1, x2, x3)) = (4)x_1   
POL(a__U32(x1, x2)) = (4)x_1   
POL(A__U112(x1, x2, x3)) = 1 + x_2 + x_3   
POL(a__U11(x1, x2)) = (2)x_1   
POL(U135(x1, x2, x3, x4)) = x_1 + (2)x_2 + x_3 + x_4   
POL(U12(x1, x2)) = (2)x_1   
POL(a__U95(x1, x2)) = x_1   
POL(U133(x1, x2, x3, x4)) = (4)x_1 + (2)x_2 + x_3 + x_4   
POL(U23(x1)) = (4)x_1   
POL(a__U131(x1, x2, x3, x4)) = (2)x_1 + (2)x_2 + x_3 + (2)x_4   
POL(a__length(x1)) = 1 + x_1   
POL(a__U102(x1, x2, x3)) = x_1   
POL(U93(x1, x2, x3)) = (2)x_1   
POL(U105(x1, x2)) = (4)x_1   
POL(a__U106(x1)) = (4)x_1   
POL(a__U13(x1)) = (4)x_1   
POL(A__U133(x1, x2, x3, x4)) = x_1 + x_4   
POL(a__U62(x1)) = (4)x_1   
POL(U31(x1, x2)) = x_1   
POL(U102(x1, x2, x3)) = x_1   
POL(U131(x1, x2, x3, x4)) = (2)x_1 + (2)x_2 + x_3 + (2)x_4   
POL(U114(x1, x2)) = 1 + x_1 + x_2   
POL(A__U135(x1, x2, x3, x4)) = x_4   
POL(a__U135(x1, x2, x3, x4)) = x_1 + (2)x_2 + x_3 + x_4   
POL(U132(x1, x2, x3, x4)) = x_1 + (2)x_2 + x_3 + (2)x_4   
POL(U62(x1)) = (4)x_1   
POL(a__U111(x1, x2, x3)) = 1 + (4)x_1 + x_2 + x_3   
POL(A__U131(x1, x2, x3, x4)) = x_2 + (2)x_4   
POL(U44(x1, x2, x3)) = x_1   
POL(a__U45(x1, x2)) = (2)x_1   
POL(U121(x1, x2)) = (4)x_1 + x_2   
POL(U33(x1)) = x_1   
POL(a__U136(x1, x2, x3, x4)) = x_1 + (2)x_2 + x_3 + x_4   
POL(U111(x1, x2, x3)) = 1 + (4)x_1 + x_2 + x_3   
POL(A__U113(x1, x2, x3)) = 1 + x_2 + x_3   
POL(a__U133(x1, x2, x3, x4)) = (4)x_1 + (2)x_2 + x_3 + x_4   
POL(U94(x1, x2, x3)) = x_1   
POL(U134(x1, x2, x3, x4)) = x_1 + (2)x_2 + x_3 + x_4   
POL(A__U114(x1, x2)) = 1 + x_1 + x_2   
POL(a__U132(x1, x2, x3, x4)) = x_1 + (2)x_2 + x_3 + (2)x_4   
POL(U21(x1, x2)) = x_1   
POL(isNatIListKind(x1)) = 0   
POL(a__U23(x1)) = (4)x_1   
POL(U136(x1, x2, x3, x4)) = x_1 + (2)x_2 + x_3 + x_4   
POL(U42(x1, x2, x3)) = (4)x_1   
POL(zeros) = 0   
POL(s(x1)) = x_1   
POL(U45(x1, x2)) = (2)x_1   
POL(a__U96(x1)) = x_1   
POL(isNat(x1)) = 0   
POL(a__isNatList(x1)) = 0   
POL(A__LENGTH(x1)) = 1 + x_1   
POL(a__U94(x1, x2, x3)) = x_1   
POL(a__U93(x1, x2, x3)) = (2)x_1   
POL(a__U44(x1, x2, x3)) = x_1   
POL(U11(x1, x2)) = (2)x_1   
POL(0) = 0   
POL(cons(x1, x2)) = x_1 + x_2   
POL(a__U42(x1, x2, x3)) = (4)x_1   
POL(a__U12(x1, x2)) = (2)x_1   
POL(a__isNatIListKind(x1)) = 0   
POL(a__U112(x1, x2, x3)) = 1 + (4)x_1 + x_2 + x_3   
POL(U96(x1)) = x_1   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U106(X)) → a__U106(mark(X))
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(isNatList(X)) → a__isNatList(X)
mark(U13(X)) → a__U13(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U52(X)) → a__U52(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U81(X)) → a__U81(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(tt) → tt
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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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))
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
mark(nil) → nil
a__zeros → zeros
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U96(X) → U96(X)
a__U95(X1, X2) → U95(X1, X2)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U106(tt) → tt
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__zeros → cons(0, zeros)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U13(tt) → tt
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__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U81(tt) → tt
a__U71(tt) → tt
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__U96(tt) → tt
a__isNat(0) → tt
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(cons(X1, X2)) → MARK(X1)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X1)
MARK(U71(X)) → MARK(X)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U114(X1, X2)) → A__U114(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
A__U114(tt, L) → A__LENGTH(mark(L))
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U96(X)) → MARK(X)
A__U136(tt, IL, M, N) → MARK(N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
MARK(U122(X)) → MARK(X)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U113(X1, X2, X3)) → A__U113(mark(X1), X2, X3)
MARK(U111(X1, X2, X3)) → A__U111(mark(X1), X2, X3)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U112(X1, X2, X3)) → A__U112(mark(X1), X2, X3)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U81(X)) → MARK(X)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                      ↳ QDPOrderProof
                    ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, L) → A__LENGTH(mark(L))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
The remaining pairs can at least be oriented weakly.

A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, L) → A__LENGTH(mark(L))
Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x1)) = x_1   
POL(U81(x1)) = 4   
POL(a__zeros) = 0   
POL(U41(x1, x2, x3)) = 2 + (3)x_3   
POL(U32(x1, x2)) = (2)x_2   
POL(mark(x1)) = x_1   
POL(a__isNatIList(x1)) = 2 + (2)x_1   
POL(U103(x1, x2, x3)) = x_2   
POL(take(x1, x2)) = x_1   
POL(a__U105(x1, x2)) = x_1   
POL(U51(x1, x2)) = 2   
POL(a__U22(x1, x2)) = x_2   
POL(U101(x1, x2, x3)) = x_2   
POL(U106(x1)) = 2   
POL(a__U46(x1)) = 2   
POL(tt) = 2   
POL(a__U33(x1)) = x_1   
POL(a__U21(x1, x2)) = x_2   
POL(U113(x1, x2, x3)) = (3)x_2   
POL(nil) = 2   
POL(a__U134(x1, x2, x3, x4)) = (3)x_3   
POL(a__U103(x1, x2, x3)) = x_2   
POL(U22(x1, x2)) = x_2   
POL(isNatKind(x1)) = 4   
POL(a__U104(x1, x2, x3)) = x_2   
POL(A__U111(x1, x2, x3)) = (4)x_1 + (2)x_2   
POL(a__U31(x1, x2)) = (2)x_2   
POL(a__U113(x1, x2, x3)) = (3)x_2   
POL(length(x1)) = x_1   
POL(U95(x1, x2)) = x_2   
POL(U112(x1, x2, x3)) = (3)x_2   
POL(a__U91(x1, x2, x3)) = (2)x_3   
POL(U43(x1, x2, x3)) = 2   
POL(U122(x1)) = 2   
POL(U104(x1, x2, x3)) = x_2   
POL(a__U101(x1, x2, x3)) = x_2   
POL(a__U121(x1, x2)) = 2   
POL(isNatList(x1)) = x_1   
POL(U52(x1)) = x_1   
POL(isNatIList(x1)) = 2 + (2)x_1   
POL(a__isNat(x1)) = x_1   
POL(U71(x1)) = 4   
POL(a__U71(x1)) = 4   
POL(a__U51(x1, x2)) = 2   
POL(a__U43(x1, x2, x3)) = 2   
POL(a__U61(x1, x2)) = 2   
POL(a__U114(x1, x2)) = (3)x_2   
POL(a__U122(x1)) = 2   
POL(a__take(x1, x2)) = x_1   
POL(a__U81(x1)) = 4   
POL(a__U41(x1, x2, x3)) = 2 + (3)x_3   
POL(U91(x1, x2, x3)) = (2)x_3   
POL(a__isNatKind(x1)) = 4   
POL(U61(x1, x2)) = 2   
POL(U46(x1)) = 2   
POL(U13(x1)) = x_1   
POL(U92(x1, x2, x3)) = x_3   
POL(a__U92(x1, x2, x3)) = x_3   
POL(a__U32(x1, x2)) = (2)x_2   
POL(A__U112(x1, x2, x3)) = (2)x_2   
POL(a__U11(x1, x2)) = x_2   
POL(U135(x1, x2, x3, x4)) = (3)x_3   
POL(U12(x1, x2)) = x_2   
POL(a__U95(x1, x2)) = x_2   
POL(U133(x1, x2, x3, x4)) = (3)x_3   
POL(U23(x1)) = x_1   
POL(a__U131(x1, x2, x3, x4)) = (3)x_3   
POL(a__length(x1)) = x_1   
POL(a__U102(x1, x2, x3)) = x_2   
POL(U93(x1, x2, x3)) = x_3   
POL(a__U13(x1)) = x_1   
POL(a__U106(x1)) = 2   
POL(U105(x1, x2)) = x_1   
POL(a__U62(x1)) = 2   
POL(U31(x1, x2)) = (2)x_2   
POL(U102(x1, x2, x3)) = x_2   
POL(U131(x1, x2, x3, x4)) = (3)x_3   
POL(U114(x1, x2)) = (3)x_2   
POL(a__U135(x1, x2, x3, x4)) = (3)x_3   
POL(U132(x1, x2, x3, x4)) = (3)x_3   
POL(U62(x1)) = 2   
POL(a__U111(x1, x2, x3)) = (3)x_2   
POL(U44(x1, x2, x3)) = x_1   
POL(a__U45(x1, x2)) = 2   
POL(U33(x1)) = x_1   
POL(U121(x1, x2)) = 2   
POL(a__U136(x1, x2, x3, x4)) = (3)x_3   
POL(U111(x1, x2, x3)) = (3)x_2   
POL(A__U113(x1, x2, x3)) = (2)x_2   
POL(a__U133(x1, x2, x3, x4)) = (3)x_3   
POL(U94(x1, x2, x3)) = x_3   
POL(U134(x1, x2, x3, x4)) = (3)x_3   
POL(A__U114(x1, x2)) = (2)x_2   
POL(a__U132(x1, x2, x3, x4)) = (3)x_3   
POL(isNatIListKind(x1)) = 2   
POL(U21(x1, x2)) = x_2   
POL(a__U23(x1)) = x_1   
POL(U136(x1, x2, x3, x4)) = (3)x_3   
POL(U42(x1, x2, x3)) = 2   
POL(zeros) = 0   
POL(s(x1)) = (3)x_1   
POL(U45(x1, x2)) = 2   
POL(a__U96(x1)) = x_1   
POL(isNat(x1)) = x_1   
POL(a__isNatList(x1)) = x_1   
POL(A__LENGTH(x1)) = (2)x_1   
POL(a__U94(x1, x2, x3)) = x_3   
POL(a__U93(x1, x2, x3)) = x_3   
POL(a__U44(x1, x2, x3)) = x_1   
POL(U11(x1, x2)) = x_2   
POL(0) = 2   
POL(cons(x1, x2)) = (3)x_2   
POL(a__U42(x1, x2, x3)) = 2   
POL(a__U12(x1, x2)) = x_2   
POL(a__isNatIListKind(x1)) = 2   
POL(U96(x1)) = x_1   
POL(a__U112(x1, x2, x3)) = (3)x_2   
The value of delta used in the strict ordering is 8.
The following usable rules [17] were oriented:

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U106(X)) → a__U106(mark(X))
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(isNatList(X)) → a__isNatList(X)
mark(U13(X)) → a__U13(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U52(X)) → a__U52(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U81(X)) → a__U81(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(tt) → tt
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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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))
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
mark(nil) → nil
a__zeros → zeros
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U96(X) → U96(X)
a__U95(X1, X2) → U95(X1, X2)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U106(tt) → tt
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__zeros → cons(0, zeros)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U13(tt) → tt
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__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U81(tt) → tt
a__U71(tt) → tt
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__U96(tt) → tt
a__isNat(0) → tt
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                      ↳ QDPOrderProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                    ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, L) → A__LENGTH(mark(L))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 4 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X1)
MARK(U71(X)) → MARK(X)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U96(X)) → MARK(X)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U136(tt, IL, M, N) → MARK(N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
MARK(U122(X)) → MARK(X)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U122(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(U71(X)) → MARK(X)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U96(X)) → MARK(X)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U136(tt, IL, M, N) → MARK(N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)
Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x1)) = x_1   
POL(a__zeros) = 3   
POL(U41(x1, x2, x3)) = (4)x_1   
POL(U81(x1)) = (4)x_1   
POL(U32(x1, x2)) = x_1   
POL(mark(x1)) = (4)x_1   
POL(a__isNatIList(x1)) = 0   
POL(take(x1, x2)) = 4 + x_1 + (4)x_2   
POL(U103(x1, x2, x3)) = (2)x_1   
POL(a__U105(x1, x2)) = x_1   
POL(A__U132(x1, x2, x3, x4)) = (2)x_4   
POL(U51(x1, x2)) = (2)x_1   
POL(a__U22(x1, x2)) = (4)x_1   
POL(U101(x1, x2, x3)) = x_1   
POL(U106(x1)) = (4)x_1   
POL(a__U46(x1)) = x_1   
POL(tt) = 0   
POL(a__U33(x1)) = x_1   
POL(a__U21(x1, x2)) = (4)x_1   
POL(U113(x1, x2, x3)) = 0   
POL(nil) = 1   
POL(a__U134(x1, x2, x3, x4)) = (4)x_1 + (4)x_4   
POL(a__U103(x1, x2, x3)) = (2)x_1   
POL(U22(x1, x2)) = (4)x_1   
POL(isNatKind(x1)) = 0   
POL(a__U104(x1, x2, x3)) = x_1   
POL(A__TAKE(x1, x2)) = 3 + (2)x_2   
POL(a__U31(x1, x2)) = x_1   
POL(a__U113(x1, x2, x3)) = 0   
POL(A__U136(x1, x2, x3, x4)) = (2)x_4   
POL(length(x1)) = 0   
POL(U95(x1, x2)) = (2)x_1   
POL(U112(x1, x2, x3)) = 0   
POL(a__U91(x1, x2, x3)) = x_1   
POL(U43(x1, x2, x3)) = x_1   
POL(U122(x1)) = 1 + (4)x_1   
POL(U104(x1, x2, x3)) = x_1   
POL(a__U101(x1, x2, x3)) = x_1   
POL(a__U121(x1, x2)) = 1 + x_1 + (4)x_2   
POL(isNatList(x1)) = 0   
POL(U52(x1)) = x_1   
POL(isNatIList(x1)) = 0   
POL(a__isNat(x1)) = 0   
POL(U71(x1)) = (2)x_1   
POL(a__U51(x1, x2)) = (2)x_1   
POL(a__U71(x1)) = (2)x_1   
POL(a__U43(x1, x2, x3)) = x_1   
POL(a__U61(x1, x2)) = (2)x_1   
POL(a__U114(x1, x2)) = 0   
POL(a__U122(x1)) = 1 + (4)x_1   
POL(a__take(x1, x2)) = 4 + x_1 + (4)x_2   
POL(a__U81(x1)) = (4)x_1   
POL(a__U41(x1, x2, x3)) = (4)x_1   
POL(U91(x1, x2, x3)) = x_1   
POL(A__U134(x1, x2, x3, x4)) = (2)x_1 + (2)x_4   
POL(MARK(x1)) = (2)x_1   
POL(a__isNatKind(x1)) = 0   
POL(U61(x1, x2)) = (2)x_1   
POL(U46(x1)) = x_1   
POL(U13(x1)) = (4)x_1   
POL(U92(x1, x2, x3)) = (4)x_1   
POL(a__U92(x1, x2, x3)) = (4)x_1   
POL(a__U32(x1, x2)) = x_1   
POL(a__U11(x1, x2)) = x_1   
POL(U135(x1, x2, x3, x4)) = (2)x_1 + x_4   
POL(U12(x1, x2)) = x_1   
POL(a__U95(x1, x2)) = (2)x_1   
POL(U133(x1, x2, x3, x4)) = x_1 + (4)x_4   
POL(U23(x1)) = x_1   
POL(a__U131(x1, x2, x3, x4)) = x_1 + (4)x_4   
POL(a__length(x1)) = 0   
POL(a__U102(x1, x2, x3)) = x_1   
POL(U93(x1, x2, x3)) = x_1   
POL(U105(x1, x2)) = x_1   
POL(a__U106(x1)) = (4)x_1   
POL(a__U13(x1)) = (4)x_1   
POL(A__U133(x1, x2, x3, x4)) = (2)x_4   
POL(a__U62(x1)) = x_1   
POL(U31(x1, x2)) = x_1   
POL(U102(x1, x2, x3)) = x_1   
POL(U131(x1, x2, x3, x4)) = x_1 + (4)x_4   
POL(A__U135(x1, x2, x3, x4)) = (2)x_4   
POL(U114(x1, x2)) = 0   
POL(a__U135(x1, x2, x3, x4)) = (2)x_1 + (4)x_4   
POL(U132(x1, x2, x3, x4)) = x_1 + (4)x_4   
POL(U62(x1)) = x_1   
POL(a__U111(x1, x2, x3)) = (2)x_1   
POL(A__U131(x1, x2, x3, x4)) = (2)x_4   
POL(U44(x1, x2, x3)) = (4)x_1   
POL(a__U45(x1, x2)) = x_1   
POL(U121(x1, x2)) = 1 + x_1 + (4)x_2   
POL(U33(x1)) = x_1   
POL(a__U136(x1, x2, x3, x4)) = (2)x_1 + (4)x_4   
POL(U111(x1, x2, x3)) = (2)x_1   
POL(a__U133(x1, x2, x3, x4)) = x_1 + (4)x_4   
POL(U94(x1, x2, x3)) = x_1   
POL(U134(x1, x2, x3, x4)) = (4)x_1 + (2)x_4   
POL(isNatIListKind(x1)) = 0   
POL(a__U132(x1, x2, x3, x4)) = x_1 + (4)x_4   
POL(U21(x1, x2)) = (4)x_1   
POL(a__U23(x1)) = x_1   
POL(U136(x1, x2, x3, x4)) = (2)x_1 + (4)x_4   
POL(U42(x1, x2, x3)) = (4)x_1   
POL(zeros) = 1   
POL(U45(x1, x2)) = x_1   
POL(s(x1)) = x_1   
POL(a__U96(x1)) = (2)x_1   
POL(isNat(x1)) = 0   
POL(a__isNatList(x1)) = 0   
POL(a__U94(x1, x2, x3)) = x_1   
POL(a__U93(x1, x2, x3)) = x_1   
POL(a__U44(x1, x2, x3)) = (4)x_1   
POL(U11(x1, x2)) = x_1   
POL(0) = 0   
POL(cons(x1, x2)) = x_1   
POL(a__U42(x1, x2, x3)) = (4)x_1   
POL(a__U12(x1, x2)) = x_1   
POL(a__isNatIListKind(x1)) = 0   
POL(a__U112(x1, x2, x3)) = 0   
POL(U96(x1)) = (2)x_1   
The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented:

mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U106(X)) → a__U106(mark(X))
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(isNatList(X)) → a__isNatList(X)
mark(U13(X)) → a__U13(mark(X))
mark(length(X)) → a__length(mark(X))
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U52(X)) → a__U52(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U81(X)) → a__U81(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(tt) → tt
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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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))
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
mark(nil) → nil
a__zeros → zeros
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U96(X) → U96(X)
a__U95(X1, X2) → U95(X1, X2)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U106(tt) → tt
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__zeros → cons(0, zeros)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U13(tt) → tt
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__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U81(tt) → tt
a__U71(tt) → tt
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__U96(tt) → tt
a__isNat(0) → tt
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                      ↳ QDPOrderProof
                        ↳ QDP
                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(U71(X)) → MARK(X)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
MARK(U96(X)) → MARK(X)
A__U136(tt, IL, M, N) → MARK(N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U11(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(U71(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U96(X)) → MARK(X)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
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__U136(tt, IL, M, N) → MARK(N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
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)
Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x1)) = 3   
POL(a__zeros) = 2   
POL(U41(x1, x2, x3)) = 1 + (4)x_1 + (3)x_2 + (3)x_3   
POL(U81(x1)) = 4 + (4)x_1   
POL(U32(x1, x2)) = 1 + (4)x_1 + (3)x_2   
POL(mark(x1)) = 3   
POL(a__isNatIList(x1)) = 1 + (2)x_1   
POL(take(x1, x2)) = 2 + (3)x_1 + x_2   
POL(U103(x1, x2, x3)) = 3 + (4)x_1 + (3)x_2 + (3)x_3   
POL(a__U105(x1, x2)) = 1 + (2)x_1 + (3)x_2   
POL(A__U132(x1, x2, x3, x4)) = 4 + (4)x_2 + (3)x_3 + (4)x_4   
POL(U51(x1, x2)) = 3 + (4)x_1 + (2)x_2   
POL(a__U22(x1, x2)) = 3 + (3)x_2   
POL(U101(x1, x2, x3)) = 3 + (4)x_1 + (3)x_2 + (3)x_3   
POL(U106(x1)) = 1 + (4)x_1   
POL(a__U46(x1)) = 4 + (3)x_1   
POL(tt) = 3   
POL(a__U33(x1)) = 3   
POL(a__U21(x1, x2)) = 2 + (3)x_2   
POL(U113(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(nil) = 0   
POL(a__U134(x1, x2, x3, x4)) = 3 + (3)x_2 + (3)x_3 + x_4   
POL(a__U103(x1, x2, x3)) = 2 + (3)x_1 + (3)x_2 + (3)x_3   
POL(U22(x1, x2)) = 3 + (4)x_1 + (3)x_2   
POL(isNatKind(x1)) = 3 + x_1   
POL(a__U104(x1, x2, x3)) = 2 + (4)x_1 + (3)x_2 + x_3   
POL(a__U31(x1, x2)) = 3 + (3)x_2   
POL(a__U113(x1, x2, x3)) = 3 + (3)x_2 + (3)x_3   
POL(A__U136(x1, x2, x3, x4)) = 3 + (3)x_2 + (3)x_3 + x_4   
POL(length(x1)) = 3 + (2)x_1   
POL(U95(x1, x2)) = 3 + (4)x_1 + (2)x_2   
POL(U112(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U91(x1, x2, x3)) = 2 + (3)x_1 + (3)x_2 + (3)x_3   
POL(U43(x1, x2, x3)) = 3 + (4)x_1 + (2)x_2 + (4)x_3   
POL(U122(x1)) = 3 + (3)x_1   
POL(U104(x1, x2, x3)) = 1 + (4)x_1 + (4)x_2 + x_3   
POL(a__U101(x1, x2, x3)) = 3 + x_2 + (3)x_3   
POL(a__U121(x1, x2)) = 3 + x_1 + (4)x_2   
POL(isNatList(x1)) = 2 + (3)x_1   
POL(U52(x1)) = 1 + (4)x_1   
POL(isNatIList(x1)) = 4 + (3)x_1   
POL(a__isNat(x1)) = 3 + x_1   
POL(U71(x1)) = 1 + (4)x_1   
POL(a__U51(x1, x2)) = 3 + (3)x_2   
POL(a__U71(x1)) = 3   
POL(a__U43(x1, x2, x3)) = 4 + (3)x_2 + (4)x_3   
POL(a__U61(x1, x2)) = 3 + (3)x_2   
POL(a__U114(x1, x2)) = 1 + x_1 + x_2   
POL(a__U122(x1)) = 2 + (3)x_1   
POL(a__take(x1, x2)) = 2 + (4)x_1 + (3)x_2   
POL(a__U81(x1)) = 3 + (3)x_1   
POL(a__U41(x1, x2, x3)) = 1 + x_1 + (3)x_2 + (3)x_3   
POL(U91(x1, x2, x3)) = 3 + x_1 + (2)x_2 + (4)x_3   
POL(A__U134(x1, x2, x3, x4)) = 4 + (4)x_2 + (3)x_3 + (4)x_4   
POL(MARK(x1)) = 3 + x_1   
POL(a__isNatKind(x1)) = 3   
POL(U61(x1, x2)) = 3 + (4)x_1 + (3)x_2   
POL(U46(x1)) = 3 + (2)x_1   
POL(U13(x1)) = 4 + (4)x_1   
POL(U92(x1, x2, x3)) = 3 + (4)x_1 + (3)x_2 + (3)x_3   
POL(a__U92(x1, x2, x3)) = 3 + (3)x_2 + (3)x_3   
POL(a__U32(x1, x2)) = 3 + (3)x_2   
POL(a__U11(x1, x2)) = 3 + (2)x_1 + (3)x_2   
POL(U135(x1, x2, x3, x4)) = 3 + x_1 + (4)x_2 + (4)x_3 + (4)x_4   
POL(U12(x1, x2)) = 3 + (4)x_1 + (2)x_2   
POL(a__U95(x1, x2)) = 4 + (3)x_1 + (3)x_2   
POL(U133(x1, x2, x3, x4)) = 1 + (4)x_1 + (4)x_2 + (4)x_3 + (4)x_4   
POL(U23(x1)) = 3 + (4)x_1   
POL(a__U131(x1, x2, x3, x4)) = 3 + (3)x_2 + (3)x_3 + (3)x_4   
POL(a__length(x1)) = 3 + (3)x_1   
POL(a__U102(x1, x2, x3)) = 2 + (2)x_2 + (3)x_3   
POL(U93(x1, x2, x3)) = 4 + (4)x_1 + (3)x_2 + (3)x_3   
POL(U105(x1, x2)) = 1 + (4)x_1 + (3)x_2   
POL(a__U106(x1)) = 2 + (2)x_1   
POL(a__U13(x1)) = 4   
POL(A__U133(x1, x2, x3, x4)) = 4 + (4)x_2 + (3)x_3 + (4)x_4   
POL(a__U62(x1)) = 1   
POL(U31(x1, x2)) = 4 + x_1 + (4)x_2   
POL(U102(x1, x2, x3)) = 1 + (4)x_1 + (2)x_2 + (3)x_3   
POL(U131(x1, x2, x3, x4)) = 1 + (3)x_1 + (4)x_2 + (4)x_3 + (4)x_4   
POL(A__U135(x1, x2, x3, x4)) = 4 + (3)x_2 + (3)x_3 + (4)x_4   
POL(U114(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(a__U135(x1, x2, x3, x4)) = 2 + (2)x_2 + x_3 + (3)x_4   
POL(U132(x1, x2, x3, x4)) = 3 + x_1 + (4)x_2 + (4)x_3 + (4)x_4   
POL(U62(x1)) = 4 + x_1   
POL(a__U111(x1, x2, x3)) = 3 + (3)x_1 + (4)x_2 + (3)x_3   
POL(A__U131(x1, x2, x3, x4)) = 4 + (4)x_2 + (3)x_3 + (4)x_4   
POL(U44(x1, x2, x3)) = 3 + (4)x_1 + (3)x_2 + (3)x_3   
POL(a__U45(x1, x2)) = 3 + (3)x_1 + (4)x_2   
POL(U33(x1)) = 1 + (4)x_1   
POL(U121(x1, x2)) = 3 + (3)x_1 + (3)x_2   
POL(a__U136(x1, x2, x3, x4)) = 1 + x_1 + (2)x_2 + (3)x_3 + (2)x_4   
POL(U111(x1, x2, x3)) = 2 + (3)x_1 + (3)x_2 + (4)x_3   
POL(a__U133(x1, x2, x3, x4)) = 3 + (3)x_2 + (3)x_3 + (3)x_4   
POL(U94(x1, x2, x3)) = 3 + (4)x_1 + (3)x_2 + (3)x_3   
POL(U134(x1, x2, x3, x4)) = 3 + (4)x_1 + (4)x_2 + (3)x_3 + (4)x_4   
POL(isNatIListKind(x1)) = 2 + (4)x_1   
POL(a__U132(x1, x2, x3, x4)) = 1 + (3)x_2 + x_3 + x_4   
POL(U21(x1, x2)) = 3 + x_1 + (2)x_2   
POL(a__U23(x1)) = 3   
POL(U136(x1, x2, x3, x4)) = 3 + (4)x_1 + (3)x_2 + (4)x_3 + (4)x_4   
POL(U42(x1, x2, x3)) = 3 + x_1 + (2)x_2 + (3)x_3   
POL(zeros) = 0   
POL(U45(x1, x2)) = 1 + (4)x_1 + (3)x_2   
POL(s(x1)) = 4 + (4)x_1   
POL(a__U96(x1)) = 3   
POL(isNat(x1)) = 3 + (4)x_1   
POL(a__isNatList(x1)) = 4   
POL(a__U94(x1, x2, x3)) = 3 + x_2 + (4)x_3   
POL(a__U93(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3   
POL(a__U44(x1, x2, x3)) = 3 + (3)x_2 + (3)x_3   
POL(U11(x1, x2)) = 3 + (2)x_1 + (3)x_2   
POL(0) = 0   
POL(cons(x1, x2)) = 3 + (4)x_1 + (3)x_2   
POL(a__U42(x1, x2, x3)) = 3 + (2)x_1 + (3)x_2 + (3)x_3   
POL(a__U12(x1, x2)) = 3   
POL(a__isNatIListKind(x1)) = 1 + (4)x_1   
POL(a__U112(x1, x2, x3)) = 3 + (3)x_2 + (3)x_3   
POL(U96(x1)) = 3 + (4)x_1   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                      ↳ QDPOrderProof
                        ↳ QDP
                          ↳ QDPOrderProof
                            ↳ QDP
                              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
A__U136(tt, IL, M, N) → MARK(N)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
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__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
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) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
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__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
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__U96(a__isNatList(V2))
a__U96(tt) → tt
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__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
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)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(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(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(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(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 7 less nodes.