The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0, IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
cons(mark(X1), X2) →⁺ mark(cons(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, cons, U102, isNatKind, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
cons < active
U102 < active
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
cons < proper
U102 < proper
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, U102, isNatKind, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
cons < active
U102 < active
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
cons < proper
U102 < proper
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol cons.

(10) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U102, active, isNatKind, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U102 < active
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U102 < proper
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U102.

(12) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
isNatKind, active, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNatKind.

(14) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U103, active, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U103.

(16) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
isNatIListKind, active, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNatIListKind.

(18) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U104, active, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U104.

(20) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U105, active, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U105.

(22) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
isNat, active, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNat.

(24) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U106, active, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U106.

(26) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
isNatIList, active, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNatIList.

(28) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U12, active, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U12.

(30) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U112, active, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U112.

(32) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U113, active, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(33) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U113.

(34) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U114, active, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(35) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U114.

(36) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
s, active, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(37) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol s.

(38) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
length, active, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(39) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol length.

(40) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U13, active, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(41) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U13.

(42) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
isNatList, active, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(43) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol isNatList.

(44) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U122, active, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(45) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U122.

(46) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U132, active, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(47) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U132.

(48) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U133, active, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(49) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U133.

(50) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U134, active, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(51) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U134.

(52) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U135, active, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(53) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U135.

(54) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U136, active, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(55) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U136.

(56) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
take, active, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(57) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol take.

(58) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U22, active, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(59) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U22.

(60) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U23, active, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(61) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U23.

(62) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U32, active, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(63) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U32.

(64) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U33, active, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(65) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U33.

(66) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U42, active, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(67) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U42.

(68) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U43, active, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(69) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U43.

(70) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U44, active, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(71) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U44.

(72) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U45, active, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(73) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U45.

(74) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U46, active, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(75) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U46.

(76) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U52, active, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(77) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U52.

(78) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U62, active, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(79) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U62.

(80) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U92, active, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(81) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U92.

(82) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U93, active, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(83) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U93.

(84) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U94, active, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(85) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U94.

(86) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U95, active, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(87) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U95.

(88) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U96, active, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(89) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U96.

(90) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U11, active, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(91) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U11.

(92) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U21, active, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(93) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U21.

(94) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U31, active, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(95) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U31.

(96) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U41, active, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(97) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U41.

(98) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U51, active, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(99) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U51.

(100) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U61, active, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(101) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U61.

(102) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U71, active, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(103) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U71.

(104) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U81, active, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(105) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U81.

(106) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U91, active, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(107) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U91.

(108) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U101, active, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

(109) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U101.

(110) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U111, active, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U111 < active
U121 < active
U131 < active
active < top
U111 < proper
U121 < proper
U131 < proper
proper < top

(111) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U111.

(112) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U121, active, U131, proper, top

They will be analysed ascendingly in the following order:
U121 < active
U131 < active
active < top
U121 < proper
U131 < proper
proper < top

(113) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U121.

(114) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
U131, active, proper, top

They will be analysed ascendingly in the following order:
U131 < active
active < top
U131 < proper
proper < top

(115) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U131.

(116) Obligation:

TRS:
Rules:
active(zeros) → mark(cons(0', zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0')) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0')
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0', IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U101 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U102 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U103 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U104 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U106 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatIList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U114 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U13 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
isNatList :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U121 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U132 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U133 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U134 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U135 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U136 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U23 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U32 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U33 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U61 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U62 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U92 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
U96 :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok → zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok → top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat → zeros:0':mark:tt:nil:ok

Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) ⇔ zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) ⇔ mark(gen_zeros:0':mark:tt:nil:ok3_0(x))

The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top