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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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')) → mark'(tt')
active'(U21'(tt')) → mark'(tt')
active'(U31'(tt')) → mark'(tt')
active'(U41'(tt', V2)) → mark'(U42'(isNatIList'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isNatList'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt', L, N)) → mark'(U62'(isNat'(N), L))
active'(U62'(tt', L)) → mark'(s'(length'(L)))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(U11'(isNatList'(V1)))
active'(isNat'(s'(V1))) → mark'(U21'(isNat'(V1)))
active'(isNatIList'(V)) → mark'(U31'(isNatList'(V)))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(U41'(isNat'(V1), V2))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(U51'(isNat'(V1), V2))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U61'(isNatList'(L), L, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X)) → U11'(active'(X))
active'(U21'(X)) → U21'(active'(X))
active'(U31'(X)) → U31'(active'(X))
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X1, X2, X3)) → U61'(active'(X1), X2, X3)
active'(U62'(X1, X2)) → U62'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X)) → mark'(U11'(X))
U21'(mark'(X)) → mark'(U21'(X))
U31'(mark'(X)) → mark'(U31'(X))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X1), X2, X3) → mark'(U61'(X1, X2, X3))
U62'(mark'(X1), X2) → mark'(U62'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X)) → U11'(proper'(X))
proper'(tt') → ok'(tt')
proper'(U21'(X)) → U21'(proper'(X))
proper'(U31'(X)) → U31'(proper'(X))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(U61'(X1, X2, X3)) → U61'(proper'(X1), proper'(X2), proper'(X3))
proper'(U62'(X1, X2)) → U62'(proper'(X1), proper'(X2))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X)) → ok'(U11'(X))
U21'(ok'(X)) → ok'(U21'(X))
U31'(ok'(X)) → ok'(U31'(X))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
U61'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U61'(X1, X2, X3))
U62'(ok'(X1), ok'(X2)) → ok'(U62'(X1, X2))
isNat'(ok'(X)) → ok'(isNat'(X))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(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')) → mark'(tt')
active'(U21'(tt')) → mark'(tt')
active'(U31'(tt')) → mark'(tt')
active'(U41'(tt', V2)) → mark'(U42'(isNatIList'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isNatList'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt', L, N)) → mark'(U62'(isNat'(N), L))
active'(U62'(tt', L)) → mark'(s'(length'(L)))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(U11'(isNatList'(V1)))
active'(isNat'(s'(V1))) → mark'(U21'(isNat'(V1)))
active'(isNatIList'(V)) → mark'(U31'(isNatList'(V)))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(U41'(isNat'(V1), V2))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(U51'(isNat'(V1), V2))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U61'(isNatList'(L), L, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X)) → U11'(active'(X))
active'(U21'(X)) → U21'(active'(X))
active'(U31'(X)) → U31'(active'(X))
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X1, X2, X3)) → U61'(active'(X1), X2, X3)
active'(U62'(X1, X2)) → U62'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X)) → mark'(U11'(X))
U21'(mark'(X)) → mark'(U21'(X))
U31'(mark'(X)) → mark'(U31'(X))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X1), X2, X3) → mark'(U61'(X1, X2, X3))
U62'(mark'(X1), X2) → mark'(U62'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X)) → U11'(proper'(X))
proper'(tt') → ok'(tt')
proper'(U21'(X)) → U21'(proper'(X))
proper'(U31'(X)) → U31'(proper'(X))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(U61'(X1, X2, X3)) → U61'(proper'(X1), proper'(X2), proper'(X3))
proper'(U62'(X1, X2)) → U62'(proper'(X1), proper'(X2))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X)) → ok'(U11'(X))
U21'(ok'(X)) → ok'(U21'(X))
U31'(ok'(X)) → ok'(U31'(X))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
U61'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U61'(X1, X2, X3))
U62'(ok'(X1), ok'(X2)) → ok'(U62'(X1, X2))
isNat'(ok'(X)) → ok'(isNat'(X))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(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'
tt' :: zeros':0':mark':tt':nil':ok'
U21' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U31' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U41' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U42' :: 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'
U51' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U52' :: 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'
U61' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U62' :: 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'
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'
nil' :: 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', U42', isNatIList', U52', isNatList', U62', isNat', s', length', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
U42' < active'
isNatIList' < active'
U52' < active'
isNatList' < active'
U62' < active'
isNat' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
cons' < proper'
U42' < proper'
isNatIList' < proper'
U52' < proper'
isNatList' < proper'
U62' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < 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', U42', isNatIList', U52', isNatList', U62', isNat', s', length', U11', U21', U31', U41', U51', U61', 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:
U42', active', isNatIList', U52', isNatList', U62', isNat', s', length', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U42' < active'
isNatIList' < active'
U52' < active'
isNatList' < active'
U62' < active'
isNat' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U42' < proper'
isNatIList' < proper'
U52' < proper'
isNatList' < proper'
U62' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3551))) → _*4, rt ∈ Ω(_n3551)

Induction Base:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n3552, 1)))) →_R^Ω(1)
mark'(U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n3552)))) →_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', U52', isNatList', U62', isNat', s', length', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
isNatIList' < active'
U52' < active'
isNatList' < active'
U62' < active'
isNat' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
isNatIList' < proper'
U52' < proper'
isNatList' < proper'
U62' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < 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:
U52', active', isNatList', U62', isNat', s', length', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U52' < active'
isNatList' < active'
U62' < active'
isNat' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U52' < proper'
isNatList' < proper'
U62' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5962))) → _*4, rt ∈ Ω(_n5962)

Induction Base:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n5963, 1)))) →_R^Ω(1)
mark'(U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n5963)))) →_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', U62', isNat', s', length', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
isNatList' < active'
U62' < active'
isNat' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
isNatList' < proper'
U62' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < 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:
U62', active', isNat', s', length', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U62' < active'
isNat' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U62' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n8503)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n8503)

Induction Base:
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n8504, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b9864)) →_R^Ω(1)
mark'(U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n8504)), _gen_zeros':0':mark':tt':nil':ok'3(_b9864))) →_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:
isNat', active', s', length', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
isNat' < proper'
s' < proper'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < 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:
s', active', length', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
s' < proper'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n12937))) → _*4, rt ∈ Ω(_n12937)

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, +(_$n12938, 1)))) →_R^Ω(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n12938)))) →_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', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
length' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n15765))) → _*4, rt ∈ Ω(_n15765)

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, +(_$n15766, 1)))) →_R^Ω(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n15766)))) →_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:
U11', active', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n18717))) → _*4, rt ∈ Ω(_n18717)

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

Induction Step:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n18718, 1)))) →_R^Ω(1)
mark'(U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n18718)))) →_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', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n21793))) → _*4, rt ∈ Ω(_n21793)

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, +(_$n21794, 1)))) →_R^Ω(1)
mark'(U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n21794)))) →_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:
U31', active', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n24993))) → _*4, rt ∈ Ω(_n24993)

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

Induction Step:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n24994, 1)))) →_R^Ω(1)
mark'(U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n24994)))) →_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:
U41', active', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U41' < active'
U51' < active'
U61' < active'
active' < top'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n28317)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n28317)

Induction Base:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n28318, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b31082)) →_R^Ω(1)
mark'(U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n28318)), _gen_zeros':0':mark':tt':nil':ok'3(_b31082))) →_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)
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3551))) → _*4, rt ∈ Ω(_n3551)
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5962))) → _*4, rt ∈ Ω(_n5962)
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n8503)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n8503)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n12937))) → _*4, rt ∈ Ω(_n12937)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n15765))) → _*4, rt ∈ Ω(_n15765)
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n18717))) → _*4, rt ∈ Ω(_n18717)
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n21793))) → _*4, rt ∈ Ω(_n21793)
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n24993))) → _*4, rt ∈ Ω(_n24993)
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n28317)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n28317)

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:
U51', active', U61', proper', top'

They will be analysed ascendingly in the following order:
U51' < active'
U61' < active'
active' < top'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n34286)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n34286)

Induction Base:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n34287, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b37375)) →_R^Ω(1)
mark'(U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n34287)), _gen_zeros':0':mark':tt':nil':ok'3(_b37375))) →_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:
U61', active', proper', top'

They will be analysed ascendingly in the following order:
U61' < active'
active' < top'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n40623)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n40623)

Induction Base:
U61'(_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))

Induction Step:
U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n40624, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b45848), _gen_zeros':0':mark':tt':nil':ok'3(_c45849)) →_R^Ω(1)
mark'(U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n40624)), _gen_zeros':0':mark':tt':nil':ok'3(_b45848), _gen_zeros':0':mark':tt':nil':ok'3(_c45849))) →_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'