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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, V2) → U62(isNatIList(activate(V2)))
U62(tt) → tt
U71(tt, L, N) → U72(isNat(activate(N)), activate(L))
U72(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → U92(isNat(activate(M)), activate(IL), activate(M), activate(N))
U92(tt, IL, M, N) → U93(isNat(activate(N)), activate(IL), activate(M), activate(N))
U93(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
isNatList(n__take(V1, V2)) → U61(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(isNatList(activate(L)), activate(L), N)
take(0, IL) → U81(isNatIList(IL))
take(s(M), cons(N, IL)) → U91(isNatIList(activate(IL)), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

zeros' → cons'(0', n__zeros')
U11'(tt') → tt'
U21'(tt') → tt'
U31'(tt') → tt'
U41'(tt', V2) → U42'(isNatIList'(activate'(V2)))
U42'(tt') → tt'
U51'(tt', V2) → U52'(isNatList'(activate'(V2)))
U52'(tt') → tt'
U61'(tt', V2) → U62'(isNatIList'(activate'(V2)))
U62'(tt') → tt'
U71'(tt', L, N) → U72'(isNat'(activate'(N)), activate'(L))
U72'(tt', L) → s'(length'(activate'(L)))
U81'(tt') → nil'
U91'(tt', IL, M, N) → U92'(isNat'(activate'(M)), activate'(IL), activate'(M), activate'(N))
U92'(tt', IL, M, N) → U93'(isNat'(activate'(N)), activate'(IL), activate'(M), activate'(N))
U93'(tt', IL, M, N) → cons'(activate'(N), n__take'(activate'(M), activate'(IL)))
isNat'(n__0') → tt'
isNat'(n__length'(V1)) → U11'(isNatList'(activate'(V1)))
isNat'(n__s'(V1)) → U21'(isNat'(activate'(V1)))
isNatIList'(V) → U31'(isNatList'(activate'(V)))
isNatIList'(n__zeros') → tt'
isNatIList'(n__cons'(V1, V2)) → U41'(isNat'(activate'(V1)), activate'(V2))
isNatList'(n__nil') → tt'
isNatList'(n__cons'(V1, V2)) → U51'(isNat'(activate'(V1)), activate'(V2))
isNatList'(n__take'(V1, V2)) → U61'(isNat'(activate'(V1)), activate'(V2))
length'(nil') → 0'
length'(cons'(N, L)) → U71'(isNatList'(activate'(L)), activate'(L), N)
take'(0', IL) → U81'(isNatIList'(IL))
take'(s'(M), cons'(N, IL)) → U91'(isNatIList'(activate'(IL)), activate'(IL), M, N)
zeros' → n__zeros'
take'(X1, X2) → n__take'(X1, X2)
0' → n__0'
length'(X) → n__length'(X)
s'(X) → n__s'(X)
cons'(X1, X2) → n__cons'(X1, X2)
nil' → n__nil'
activate'(n__zeros') → zeros'
activate'(n__take'(X1, X2)) → take'(X1, X2)
activate'(n__0') → 0'
activate'(n__length'(X)) → length'(X)
activate'(n__s'(X)) → s'(X)
activate'(n__cons'(X1, X2)) → cons'(X1, X2)
activate'(n__nil') → nil'
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
zeros' → cons'(0', n__zeros')
U11'(tt') → tt'
U21'(tt') → tt'
U31'(tt') → tt'
U41'(tt', V2) → U42'(isNatIList'(activate'(V2)))
U42'(tt') → tt'
U51'(tt', V2) → U52'(isNatList'(activate'(V2)))
U52'(tt') → tt'
U61'(tt', V2) → U62'(isNatIList'(activate'(V2)))
U62'(tt') → tt'
U71'(tt', L, N) → U72'(isNat'(activate'(N)), activate'(L))
U72'(tt', L) → s'(length'(activate'(L)))
U81'(tt') → nil'
U91'(tt', IL, M, N) → U92'(isNat'(activate'(M)), activate'(IL), activate'(M), activate'(N))
U92'(tt', IL, M, N) → U93'(isNat'(activate'(N)), activate'(IL), activate'(M), activate'(N))
U93'(tt', IL, M, N) → cons'(activate'(N), n__take'(activate'(M), activate'(IL)))
isNat'(n__0') → tt'
isNat'(n__length'(V1)) → U11'(isNatList'(activate'(V1)))
isNat'(n__s'(V1)) → U21'(isNat'(activate'(V1)))
isNatIList'(V) → U31'(isNatList'(activate'(V)))
isNatIList'(n__zeros') → tt'
isNatIList'(n__cons'(V1, V2)) → U41'(isNat'(activate'(V1)), activate'(V2))
isNatList'(n__nil') → tt'
isNatList'(n__cons'(V1, V2)) → U51'(isNat'(activate'(V1)), activate'(V2))
isNatList'(n__take'(V1, V2)) → U61'(isNat'(activate'(V1)), activate'(V2))
length'(nil') → 0'
length'(cons'(N, L)) → U71'(isNatList'(activate'(L)), activate'(L), N)
take'(0', IL) → U81'(isNatIList'(IL))
take'(s'(M), cons'(N, IL)) → U91'(isNatIList'(activate'(IL)), activate'(IL), M, N)
zeros' → n__zeros'
take'(X1, X2) → n__take'(X1, X2)
0' → n__0'
length'(X) → n__length'(X)
s'(X) → n__s'(X)
cons'(X1, X2) → n__cons'(X1, X2)
nil' → n__nil'
activate'(n__zeros') → zeros'
activate'(n__take'(X1, X2)) → take'(X1, X2)
activate'(n__0') → 0'
activate'(n__length'(X)) → length'(X)
activate'(n__s'(X)) → s'(X)
activate'(n__cons'(X1, X2)) → cons'(X1, X2)
activate'(n__nil') → nil'
activate'(X) → X

Types:
zeros' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
cons' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
0' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
n__zeros' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
U11' :: tt' → tt'
tt' :: tt'
U21' :: tt' → tt'
U31' :: tt' → tt'
U41' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → tt'
U42' :: tt' → tt'
isNatIList' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → tt'
activate' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
U51' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → tt'
U52' :: tt' → tt'
isNatList' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → tt'
U61' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → tt'
U62' :: tt' → tt'
U71' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
U72' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
isNat' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → tt'
s' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
length' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
U81' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
nil' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
U91' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
U92' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
U93' :: tt' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
n__take' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
n__0' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
n__length' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
n__s' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
n__cons' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
n__nil' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
take' :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
_hole_n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'1 :: n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'
_hole_tt'2 :: tt'
_gen_n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'3 :: Nat → n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'

Heuristically decided to analyse the following defined symbols:
isNatIList', activate', isNatList', isNat', length'

They will be analysed ascendingly in the following order:
isNatIList' = activate'
isNatIList' = isNatList'
isNatIList' = isNat'
isNatIList' = length'
activate' = isNatList'
activate' = isNat'
activate' = length'
isNatList' = isNat'
isNatList' = length'
isNat' = length'

Generator Equations:
_gen_n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'3(0) ⇔ n__zeros'
_gen_n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'3(+(x, 1)) ⇔ n__take'(n__zeros', _gen_n__zeros':n__take':n__0':n__length':n__s':n__cons':n__nil'3(x))

The following defined symbols remain to be analysed:
isNatList', isNatIList', activate', isNat', length'

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

The following defined symbols remain to be analysed:
isNat', isNatIList', activate', length'

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

The following defined symbols remain to be analysed:
activate', isNatIList', length'

Could not prove a rewrite lemma for the defined symbol activate'.

The following defined symbols remain to be analysed:
length', isNatIList'

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

The following defined symbols remain to be analysed:
isNatIList'

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

No more defined symbols left to analyse.