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

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
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__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
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', V1) → U12'(isNatList'(activate'(V1)))
U12'(tt') → tt'
U21'(tt', V1) → U22'(isNat'(activate'(V1)))
U22'(tt') → tt'
U31'(tt', V) → U32'(isNatList'(activate'(V)))
U32'(tt') → tt'
U41'(tt', V1, V2) → U42'(isNat'(activate'(V1)), activate'(V2))
U42'(tt', V2) → U43'(isNatIList'(activate'(V2)))
U43'(tt') → tt'
U51'(tt', V1, V2) → U52'(isNat'(activate'(V1)), activate'(V2))
U52'(tt', V2) → U53'(isNatList'(activate'(V2)))
U53'(tt') → tt'
U61'(tt', L) → s'(length'(activate'(L)))
and'(tt', X) → activate'(X)
isNat'(n__0') → tt'
isNat'(n__length'(V1)) → U11'(isNatIListKind'(activate'(V1)), activate'(V1))
isNat'(n__s'(V1)) → U21'(isNatKind'(activate'(V1)), activate'(V1))
isNatIList'(V) → U31'(isNatIListKind'(activate'(V)), activate'(V))
isNatIList'(n__zeros') → tt'
isNatIList'(n__cons'(V1, V2)) → U41'(and'(isNatKind'(activate'(V1)), n__isNatIListKind'(activate'(V2))), activate'(V1), activate'(V2))
isNatIListKind'(n__nil') → tt'
isNatIListKind'(n__zeros') → tt'
isNatIListKind'(n__cons'(V1, V2)) → and'(isNatKind'(activate'(V1)), n__isNatIListKind'(activate'(V2)))
isNatKind'(n__0') → tt'
isNatKind'(n__length'(V1)) → isNatIListKind'(activate'(V1))
isNatKind'(n__s'(V1)) → isNatKind'(activate'(V1))
isNatList'(n__nil') → tt'
isNatList'(n__cons'(V1, V2)) → U51'(and'(isNatKind'(activate'(V1)), n__isNatIListKind'(activate'(V2))), activate'(V1), activate'(V2))
length'(nil') → 0'
length'(cons'(N, L)) → U61'(and'(and'(isNatList'(activate'(L)), n__isNatIListKind'(activate'(L))), n__and'(isNat'(N), n__isNatKind'(N))), activate'(L))
zeros' → n__zeros'
0' → n__0'
length'(X) → n__length'(X)
s'(X) → n__s'(X)
cons'(X1, X2) → n__cons'(X1, X2)
isNatIListKind'(X) → n__isNatIListKind'(X)
nil' → n__nil'
and'(X1, X2) → n__and'(X1, X2)
isNatKind'(X) → n__isNatKind'(X)
activate'(n__zeros') → zeros'
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__isNatIListKind'(X)) → isNatIListKind'(X)
activate'(n__nil') → nil'
activate'(n__and'(X1, X2)) → and'(X1, X2)
activate'(n__isNatKind'(X)) → isNatKind'(X)
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
zeros' → cons'(0', n__zeros')
U11'(tt', V1) → U12'(isNatList'(activate'(V1)))
U12'(tt') → tt'
U21'(tt', V1) → U22'(isNat'(activate'(V1)))
U22'(tt') → tt'
U31'(tt', V) → U32'(isNatList'(activate'(V)))
U32'(tt') → tt'
U41'(tt', V1, V2) → U42'(isNat'(activate'(V1)), activate'(V2))
U42'(tt', V2) → U43'(isNatIList'(activate'(V2)))
U43'(tt') → tt'
U51'(tt', V1, V2) → U52'(isNat'(activate'(V1)), activate'(V2))
U52'(tt', V2) → U53'(isNatList'(activate'(V2)))
U53'(tt') → tt'
U61'(tt', L) → s'(length'(activate'(L)))
and'(tt', X) → activate'(X)
isNat'(n__0') → tt'
isNat'(n__length'(V1)) → U11'(isNatIListKind'(activate'(V1)), activate'(V1))
isNat'(n__s'(V1)) → U21'(isNatKind'(activate'(V1)), activate'(V1))
isNatIList'(V) → U31'(isNatIListKind'(activate'(V)), activate'(V))
isNatIList'(n__zeros') → tt'
isNatIList'(n__cons'(V1, V2)) → U41'(and'(isNatKind'(activate'(V1)), n__isNatIListKind'(activate'(V2))), activate'(V1), activate'(V2))
isNatIListKind'(n__nil') → tt'
isNatIListKind'(n__zeros') → tt'
isNatIListKind'(n__cons'(V1, V2)) → and'(isNatKind'(activate'(V1)), n__isNatIListKind'(activate'(V2)))
isNatKind'(n__0') → tt'
isNatKind'(n__length'(V1)) → isNatIListKind'(activate'(V1))
isNatKind'(n__s'(V1)) → isNatKind'(activate'(V1))
isNatList'(n__nil') → tt'
isNatList'(n__cons'(V1, V2)) → U51'(and'(isNatKind'(activate'(V1)), n__isNatIListKind'(activate'(V2))), activate'(V1), activate'(V2))
length'(nil') → 0'
length'(cons'(N, L)) → U61'(and'(and'(isNatList'(activate'(L)), n__isNatIListKind'(activate'(L))), n__and'(isNat'(N), n__isNatKind'(N))), activate'(L))
zeros' → n__zeros'
0' → n__0'
length'(X) → n__length'(X)
s'(X) → n__s'(X)
cons'(X1, X2) → n__cons'(X1, X2)
isNatIListKind'(X) → n__isNatIListKind'(X)
nil' → n__nil'
and'(X1, X2) → n__and'(X1, X2)
isNatKind'(X) → n__isNatKind'(X)
activate'(n__zeros') → zeros'
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__isNatIListKind'(X)) → isNatIListKind'(X)
activate'(n__nil') → nil'
activate'(n__and'(X1, X2)) → and'(X1, X2)
activate'(n__isNatKind'(X)) → isNatKind'(X)
activate'(X) → X

Types:
zeros' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
cons' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
0' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__zeros' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U11' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
tt' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U12' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
isNatList' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
activate' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U21' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U22' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
isNat' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U31' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U32' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U41' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U42' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U43' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
isNatIList' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U51' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U52' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U53' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
U61' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
s' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
length' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
and' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__0' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__length' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
isNatIListKind' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__s' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
isNatKind' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__cons' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__isNatIListKind' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__nil' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
nil' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__and' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
n__isNatKind' :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and' → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
_hole_n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'1 :: n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'
_gen_n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'2 :: Nat → n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'

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

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

Generator Equations:
_gen_n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'2(0) ⇔ n__zeros'
_gen_n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'2(+(x, 1)) ⇔ n__length'(_gen_n__zeros':tt':n__0':n__length':n__s':n__cons':n__isNatIListKind':n__nil':n__isNatKind':n__and'2(x))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.