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:

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

Rewrite Strategy: FULL

(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) Obligation:

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

a__and(tt, T) → mark(T)
a__isNatIList(IL) → a__isNatList(IL)
a__isNat(0) → tt
a__isNat(s(N)) → a__isNat(N)
a__isNat(length(L)) → a__isNatList(L)
a__isNatIList(zeros) → tt
a__isNatIList(cons(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__isNatList(nil) → tt
a__isNatList(cons(N, L)) → a__and(a__isNat(N), a__isNatList(L))
a__isNatList(take(N, IL)) → a__and(a__isNat(N), a__isNatIList(IL))
a__zeros → cons(0, zeros)
a__take(0, IL) → a__uTake1(a__isNatIList(IL))
a__uTake1(tt) → nil
a__take(s(M), cons(N, IL)) → a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL)
a__uTake2(tt, M, N, IL) → cons(mark(N), take(M, IL))
a__length(cons(N, L)) → a__uLength(a__and(a__isNat(N), a__isNatList(L)), L)
a__uLength(tt, L) → s(a__length(mark(L)))
mark(and(X1, X2)) → a__and(mark(X1), mark(X2))
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(zeros) → a__zeros
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(uTake1(X)) → a__uTake1(mark(X))
mark(uTake2(X1, X2, X3, X4)) → a__uTake2(mark(X1), X2, X3, X4)
mark(uLength(X1, X2)) → a__uLength(mark(X1), X2)
mark(tt) → tt
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
a__and(X1, X2) → and(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__zeros → zeros
a__take(X1, X2) → take(X1, X2)
a__uTake1(X) → uTake1(X)
a__uTake2(X1, X2, X3, X4) → uTake2(X1, X2, X3, X4)
a__uLength(X1, X2) → uLength(X1, X2)

S is empty.
Rewrite Strategy: FULL

(3) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a__isNatIList(take(N1387_5, IL1388_5)) →⁺ a__and(a__isNat(N1387_5), a__isNatIList(IL1388_5))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [IL1388_5 / take(N1387_5, IL1388_5)].
The result substitution is [ ].

(4) BOUNDS(n^1, INF)