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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(U21(tt)) → mark(nil)
active(U31(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(isNatList(take(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
active(take(0, IL)) → mark(U21(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2, X3, X4)) → U31(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2, X3, X4) → mark(U31(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U21(X)) → U21(proper(X))
proper(nil) → ok(nil)
proper(U31(X1, X2, X3, X4)) → U31(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U31(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
isNatList(ok(X)) → ok(isNatList(X))
isNatIList(ok(X)) → ok(isNatIList(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

active'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt', L)) → mark'(s'(length'(L)))
active'(U21'(tt')) → mark'(nil')
active'(U31'(tt', IL, M, N)) → mark'(cons'(N, take'(M, IL)))
active'(and'(tt', X)) → mark'(X)
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(isNatList'(V1))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(isNatIList'(V)) → mark'(isNatList'(V))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(and'(isNat'(V1), isNatIList'(V2)))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(and'(isNat'(V1), isNatList'(V2)))
active'(isNatList'(take'(V1, V2))) → mark'(and'(isNat'(V1), isNatIList'(V2)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(and'(isNatList'(L), isNat'(N)), L))
active'(take'(0', IL)) → mark'(U21'(isNatIList'(IL)))
active'(take'(s'(M), cons'(N, IL))) → mark'(U31'(and'(isNatIList'(IL), and'(isNat'(M), isNat'(N))), IL, M, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(U21'(X)) → U21'(active'(X))
active'(U31'(X1, X2, X3, X4)) → U31'(active'(X1), X2, X3, X4)
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(and'(X1, X2)) → and'(active'(X1), X2)
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
U21'(mark'(X)) → mark'(U21'(X))
U31'(mark'(X1), X2, X3, X4) → mark'(U31'(X1, X2, X3, X4))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(U21'(X)) → U21'(proper'(X))
proper'(nil') → ok'(nil')
proper'(U31'(X1, X2, X3, X4)) → U31'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
U21'(ok'(X)) → ok'(U21'(X))
U31'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U31'(X1, X2, X3, X4))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isNat'(ok'(X)) → ok'(isNat'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt', L)) → mark'(s'(length'(L)))
active'(U21'(tt')) → mark'(nil')
active'(U31'(tt', IL, M, N)) → mark'(cons'(N, take'(M, IL)))
active'(and'(tt', X)) → mark'(X)
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(isNatList'(V1))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(isNatIList'(V)) → mark'(isNatList'(V))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(and'(isNat'(V1), isNatIList'(V2)))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(and'(isNat'(V1), isNatList'(V2)))
active'(isNatList'(take'(V1, V2))) → mark'(and'(isNat'(V1), isNatIList'(V2)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(and'(isNatList'(L), isNat'(N)), L))
active'(take'(0', IL)) → mark'(U21'(isNatIList'(IL)))
active'(take'(s'(M), cons'(N, IL))) → mark'(U31'(and'(isNatIList'(IL), and'(isNat'(M), isNat'(N))), IL, M, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(U21'(X)) → U21'(active'(X))
active'(U31'(X1, X2, X3, X4)) → U31'(active'(X1), X2, X3, X4)
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
active'(and'(X1, X2)) → and'(active'(X1), X2)
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
U21'(mark'(X)) → mark'(U21'(X))
U31'(mark'(X1), X2, X3, X4) → mark'(U31'(X1, X2, X3, X4))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(U21'(X)) → U21'(proper'(X))
proper'(nil') → ok'(nil')
proper'(U31'(X1, X2, X3, X4)) → U31'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
U21'(ok'(X)) → ok'(U21'(X))
U31'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U31'(X1, X2, X3, X4))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isNat'(ok'(X)) → ok'(isNat'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(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'
U11' :: 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'
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'
U21' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: 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' → 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'
and' :: 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'
isNatList' :: 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'
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':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'

Heuristically decided to analyse the following defined symbols:
active', cons', s', length', take', isNatList', isNat', and', isNatIList', U11', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
s' < active'
length' < active'
take' < active'
isNatList' < active'
isNat' < active'
and' < active'
isNatIList' < active'
U11' < active'
U21' < active'
U31' < active'
active' < top'
cons' < proper'
s' < proper'
length' < proper'
take' < proper'
isNatList' < proper'
isNat' < proper'
and' < proper'
isNatIList' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
cons', active', s', length', take', isNatList', isNat', and', isNatIList', U11', U21', U31', proper', top'

Proved the following rewrite lemma:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)

Induction Base:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n6, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b610)) →_R^Ω(1)
mark'(cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n6)), _gen_zeros':0':mark':tt':nil':ok'3(_b610))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
s', active', length', take', isNatList', isNat', and', isNatIList', U11', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
length' < active'
take' < active'
isNatList' < active'
isNat' < active'
and' < active'
isNatIList' < active'
U11' < active'
U21' < active'
U31' < active'
active' < top'
s' < proper'
length' < proper'
take' < proper'
isNatList' < proper'
isNat' < proper'
and' < proper'
isNatIList' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3317))) → _*4, rt ∈ Ω(_n3317)

Induction Base:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n3318, 1)))) →_R^Ω(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n3318)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
length', active', take', isNatList', isNat', and', isNatIList', U11', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
length' < active'
take' < active'
isNatList' < active'
isNat' < active'
and' < active'
isNatIList' < active'
U11' < active'
U21' < active'
U31' < active'
active' < top'
length' < proper'
take' < proper'
isNatList' < proper'
isNat' < proper'
and' < proper'
isNatIList' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

Proved the following rewrite lemma:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5580))) → _*4, rt ∈ Ω(_n5580)

Induction Base:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n5581, 1)))) →_R^Ω(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n5581)))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
take', active', isNatList', isNat', and', isNatIList', U11', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
take' < active'
isNatList' < active'
isNat' < active'
and' < active'
isNatIList' < active'
U11' < active'
U21' < active'
U31' < active'
active' < top'
take' < proper'
isNatList' < proper'
isNat' < proper'
and' < proper'
isNatIList' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

Proved the following rewrite lemma:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7967)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n7967)

Induction Base:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n7968, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b9544)) →_R^Ω(1)
mark'(take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n7968)), _gen_zeros':0':mark':tt':nil':ok'3(_b9544))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
isNatList', active', isNat', and', isNatIList', U11', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
isNatList' < active'
isNat' < active'
and' < active'
isNatIList' < active'
U11' < active'
U21' < active'
U31' < active'
active' < top'
isNatList' < proper'
isNat' < proper'
and' < proper'
isNatIList' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
isNat', active', and', isNatIList', U11', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
and' < active'
isNatIList' < active'
U11' < active'
U21' < active'
U31' < active'
active' < top'
isNat' < proper'
and' < proper'
isNatIList' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
and', active', isNatIList', U11', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
and' < active'
isNatIList' < active'
U11' < active'
U21' < active'
U31' < active'
active' < top'
and' < proper'
isNatIList' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

Proved the following rewrite lemma:
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n12424)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n12424)

Induction Base:
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n12425, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b14109)) →_R^Ω(1)
mark'(and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n12425)), _gen_zeros':0':mark':tt':nil':ok'3(_b14109))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
isNatIList', active', U11', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
isNatIList' < active'
U11' < active'
U21' < active'
U31' < active'
active' < top'
isNatIList' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U11', active', U21', U31', proper', top'

They will be analysed ascendingly in the following order:
U11' < active'
U21' < active'
U31' < active'
active' < top'
U11' < proper'
U21' < proper'
U31' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n17002)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n17002)

Induction Base:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n17003, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b19011)) →_R^Ω(1)
mark'(U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n17003)), _gen_zeros':0':mark':tt':nil':ok'3(_b19011))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U21', active', U31', proper', top'

They will be analysed ascendingly in the following order:
U21' < active'
U31' < active'
active' < top'
U21' < proper'
U31' < proper'
proper' < top'

Proved the following rewrite lemma:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n21900))) → _*4, rt ∈ Ω(_n21900)

Induction Base:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n21901, 1)))) →_R^Ω(1)
mark'(U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n21901)))) →_IH
mark'(_*4)

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

Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3317))) → _*4, rt ∈ Ω(_n3317)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5580))) → _*4, rt ∈ Ω(_n5580)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7967)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n7967)
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n12424)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n12424)
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n17002)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n17002)
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n21900))) → _*4, rt ∈ Ω(_n21900)

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U31', active', proper', top'

They will be analysed ascendingly in the following order:
U31' < active'
active' < top'
U31' < proper'
proper' < top'

Proved the following rewrite lemma:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n24984)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n24984)

Induction Base:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n24985, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b30509), _gen_zeros':0':mark':tt':nil':ok'3(_c30510), _gen_zeros':0':mark':tt':nil':ok'3(_d30511)) →_R^Ω(1)
mark'(U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n24985)), _gen_zeros':0':mark':tt':nil':ok'3(_b30509), _gen_zeros':0':mark':tt':nil':ok'3(_c30510), _gen_zeros':0':mark':tt':nil':ok'3(_d30511))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'