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, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
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))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
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__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(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', L, N) → U62'(isNat'(activate'(N)), activate'(L))
U62'(tt', L) → s'(length'(activate'(L)))
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))
length'(nil') → 0'
length'(cons'(N, L)) → U61'(isNatList'(activate'(L)), activate'(L), N)
zeros' → n__zeros'
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__0') → 0'
activate'(n__length'(X)) → length'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(n__cons'(X1, X2)) → cons'(activate'(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', L, N) → U62'(isNat'(activate'(N)), activate'(L))
U62'(tt', L) → s'(length'(activate'(L)))
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))
length'(nil') → 0'
length'(cons'(N, L)) → U61'(isNatList'(activate'(L)), activate'(L), N)
zeros' → n__zeros'
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__0') → 0'
activate'(n__length'(X)) → length'(activate'(X))
activate'(n__s'(X)) → s'(activate'(X))
activate'(n__cons'(X1, X2)) → cons'(activate'(X1), X2)
activate'(n__nil') → nil'
activate'(X) → X

Types:
zeros' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil'
cons' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
0' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil'
n__zeros' :: n__zeros':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__0':n__length':n__s':n__cons':n__nil' → tt'
U42' :: tt' → tt'
isNatIList' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → tt'
activate' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
U51' :: tt' → n__zeros':n__0':n__length':n__s':n__cons':n__nil' → tt'
U52' :: tt' → tt'
isNatList' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → tt'
U61' :: tt' → n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
U62' :: tt' → n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
isNat' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → tt'
s' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
length' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
n__0' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil'
n__length' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
n__s' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
n__cons' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil' → n__zeros':n__0':n__length':n__s':n__cons':n__nil'
n__nil' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil'
nil' :: n__zeros':n__0':n__length':n__s':n__cons':n__nil'
_hole_n__zeros':n__0':n__length':n__s':n__cons':n__nil'1 :: n__zeros':n__0':n__length':n__s':n__cons':n__nil'
_hole_tt'2 :: tt'
_gen_n__zeros':n__0':n__length':n__s':n__cons':n__nil'3 :: Nat → n__zeros':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:
activate' < isNatIList'
isNatList' < isNatIList'
isNat' < isNatIList'
activate' = isNatList'
activate' = isNat'
activate' = length'
isNatList' = isNat'
isNatList' = length'
isNat' = length'

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

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

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

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

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

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

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

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

Proved the following rewrite lemma:
activate'(_gen_n__zeros':n__0':n__length':n__s':n__cons':n__nil'3(+(1, _n20850))) → _*4, rt ∈ Ω(_n20850)

Induction Base:
activate'(_gen_n__zeros':n__0':n__length':n__s':n__cons':n__nil'3(+(1, 0)))

Induction Step:
activate'(_gen_n__zeros':n__0':n__length':n__s':n__cons':n__nil'3(+(1, +(_$n20851, 1)))) →_R^Ω(1)
length'(activate'(_gen_n__zeros':n__0':n__length':n__s':n__cons':n__nil'3(+(1, _$n20851)))) →_IH
length'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).