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

active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Heuristically decided to analyse the following defined symbols:
active', cons', app', from', s', zWadr', prefix', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
app' < active'
from' < active'
s' < active'
zWadr' < active'
prefix' < active'
active' < top'
cons' < proper'
app' < proper'
from' < proper'
s' < proper'
zWadr' < proper'
prefix' < proper'
proper' < top'

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

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

The following defined symbols remain to be analysed:
cons', active', app', from', s', zWadr', prefix', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
app' < active'
from' < active'
s' < active'
zWadr' < active'
prefix' < active'
active' < top'
cons' < proper'
app' < proper'
from' < proper'
s' < proper'
zWadr' < proper'
prefix' < proper'
proper' < top'

Proved the following rewrite lemma:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)

Induction Base:
cons'(_gen_nil':mark':ok'3(+(1, 0)), _gen_nil':mark':ok'3(b))

Induction Step:
cons'(_gen_nil':mark':ok'3(+(1, +(_$n6, 1))), _gen_nil':mark':ok'3(_b610)) →RΩ(1)
mark'(cons'(_gen_nil':mark':ok'3(+(1, _$n6)), _gen_nil':mark':ok'3(_b610))) →IH
mark'(_*4)

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)

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

The following defined symbols remain to be analysed:
app', active', from', s', zWadr', prefix', proper', top'

They will be analysed ascendingly in the following order:
app' < active'
from' < active'
s' < active'
zWadr' < active'
prefix' < active'
active' < top'
app' < proper'
from' < proper'
s' < proper'
zWadr' < proper'
prefix' < proper'
proper' < top'

Proved the following rewrite lemma:
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)

Induction Base:
app'(_gen_nil':mark':ok'3(+(1, 0)), _gen_nil':mark':ok'3(b))

Induction Step:
app'(_gen_nil':mark':ok'3(+(1, +(_$n2292, 1))), _gen_nil':mark':ok'3(_b3436)) →RΩ(1)
mark'(app'(_gen_nil':mark':ok'3(+(1, _$n2292)), _gen_nil':mark':ok'3(_b3436))) →IH
mark'(_*4)

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)

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

The following defined symbols remain to be analysed:
from', active', s', zWadr', prefix', proper', top'

They will be analysed ascendingly in the following order:
from' < active'
s' < active'
zWadr' < active'
prefix' < active'
active' < top'
from' < proper'
s' < proper'
zWadr' < proper'
prefix' < proper'
proper' < top'

Proved the following rewrite lemma:
from'(_gen_nil':mark':ok'3(+(1, _n5164))) → _*4, rt ∈ Ω(n5164)

Induction Base:
from'(_gen_nil':mark':ok'3(+(1, 0)))

Induction Step:
from'(_gen_nil':mark':ok'3(+(1, +(_$n5165, 1)))) →RΩ(1)
mark'(from'(_gen_nil':mark':ok'3(+(1, _$n5165)))) →IH
mark'(_*4)

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)
from'(_gen_nil':mark':ok'3(+(1, _n5164))) → _*4, rt ∈ Ω(n5164)

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

The following defined symbols remain to be analysed:
s', active', zWadr', prefix', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
zWadr' < active'
prefix' < active'
active' < top'
s' < proper'
zWadr' < proper'
prefix' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_nil':mark':ok'3(+(1, _n6952))) → _*4, rt ∈ Ω(n6952)

Induction Base:
s'(_gen_nil':mark':ok'3(+(1, 0)))

Induction Step:
s'(_gen_nil':mark':ok'3(+(1, +(_$n6953, 1)))) →RΩ(1)
mark'(s'(_gen_nil':mark':ok'3(+(1, _$n6953)))) →IH
mark'(_*4)

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)
from'(_gen_nil':mark':ok'3(+(1, _n5164))) → _*4, rt ∈ Ω(n5164)
s'(_gen_nil':mark':ok'3(+(1, _n6952))) → _*4, rt ∈ Ω(n6952)

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

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

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

Proved the following rewrite lemma:
zWadr'(_gen_nil':mark':ok'3(+(1, _n8864)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n8864)

Induction Base:
zWadr'(_gen_nil':mark':ok'3(+(1, 0)), _gen_nil':mark':ok'3(b))

Induction Step:
zWadr'(_gen_nil':mark':ok'3(+(1, +(_$n8865, 1))), _gen_nil':mark':ok'3(_b10765)) →RΩ(1)
mark'(zWadr'(_gen_nil':mark':ok'3(+(1, _$n8865)), _gen_nil':mark':ok'3(_b10765))) →IH
mark'(_*4)

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)
from'(_gen_nil':mark':ok'3(+(1, _n5164))) → _*4, rt ∈ Ω(n5164)
s'(_gen_nil':mark':ok'3(+(1, _n6952))) → _*4, rt ∈ Ω(n6952)
zWadr'(_gen_nil':mark':ok'3(+(1, _n8864)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n8864)

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

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

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

Proved the following rewrite lemma:
prefix'(_gen_nil':mark':ok'3(+(1, _n12587))) → _*4, rt ∈ Ω(n12587)

Induction Base:
prefix'(_gen_nil':mark':ok'3(+(1, 0)))

Induction Step:
prefix'(_gen_nil':mark':ok'3(+(1, +(_$n12588, 1)))) →RΩ(1)
mark'(prefix'(_gen_nil':mark':ok'3(+(1, _$n12588)))) →IH
mark'(_*4)

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)
from'(_gen_nil':mark':ok'3(+(1, _n5164))) → _*4, rt ∈ Ω(n5164)
s'(_gen_nil':mark':ok'3(+(1, _n6952))) → _*4, rt ∈ Ω(n6952)
zWadr'(_gen_nil':mark':ok'3(+(1, _n8864)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n8864)
prefix'(_gen_nil':mark':ok'3(+(1, _n12587))) → _*4, rt ∈ Ω(n12587)

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

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)
from'(_gen_nil':mark':ok'3(+(1, _n5164))) → _*4, rt ∈ Ω(n5164)
s'(_gen_nil':mark':ok'3(+(1, _n6952))) → _*4, rt ∈ Ω(n6952)
zWadr'(_gen_nil':mark':ok'3(+(1, _n8864)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n8864)
prefix'(_gen_nil':mark':ok'3(+(1, _n12587))) → _*4, rt ∈ Ω(n12587)

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

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

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

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)
from'(_gen_nil':mark':ok'3(+(1, _n5164))) → _*4, rt ∈ Ω(n5164)
s'(_gen_nil':mark':ok'3(+(1, _n6952))) → _*4, rt ∈ Ω(n6952)
zWadr'(_gen_nil':mark':ok'3(+(1, _n8864)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n8864)
prefix'(_gen_nil':mark':ok'3(+(1, _n12587))) → _*4, rt ∈ Ω(n12587)

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

The following defined symbols remain to be analysed:
top'

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

Rules:
active'(app'(nil', YS)) → mark'(YS)
active'(app'(cons'(X, XS), YS)) → mark'(cons'(X, app'(XS, YS)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(zWadr'(nil', YS)) → mark'(nil')
active'(zWadr'(XS, nil')) → mark'(nil')
active'(zWadr'(cons'(X, XS), cons'(Y, YS))) → mark'(cons'(app'(Y, cons'(X, nil')), zWadr'(XS, YS)))
active'(prefix'(L)) → mark'(cons'(nil', zWadr'(L, prefix'(L))))
active'(app'(X1, X2)) → app'(active'(X1), X2)
active'(app'(X1, X2)) → app'(X1, active'(X2))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(zWadr'(X1, X2)) → zWadr'(active'(X1), X2)
active'(zWadr'(X1, X2)) → zWadr'(X1, active'(X2))
active'(prefix'(X)) → prefix'(active'(X))
app'(mark'(X1), X2) → mark'(app'(X1, X2))
app'(X1, mark'(X2)) → mark'(app'(X1, X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
zWadr'(mark'(X1), X2) → mark'(zWadr'(X1, X2))
zWadr'(X1, mark'(X2)) → mark'(zWadr'(X1, X2))
prefix'(mark'(X)) → mark'(prefix'(X))
proper'(app'(X1, X2)) → app'(proper'(X1), proper'(X2))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(zWadr'(X1, X2)) → zWadr'(proper'(X1), proper'(X2))
proper'(prefix'(X)) → prefix'(proper'(X))
app'(ok'(X1), ok'(X2)) → ok'(app'(X1, X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
zWadr'(ok'(X1), ok'(X2)) → ok'(zWadr'(X1, X2))
prefix'(ok'(X)) → ok'(prefix'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: nil':mark':ok' → nil':mark':ok'
app' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
nil' :: nil':mark':ok'
mark' :: nil':mark':ok' → nil':mark':ok'
cons' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
from' :: nil':mark':ok' → nil':mark':ok'
s' :: nil':mark':ok' → nil':mark':ok'
zWadr' :: nil':mark':ok' → nil':mark':ok' → nil':mark':ok'
prefix' :: nil':mark':ok' → nil':mark':ok'
proper' :: nil':mark':ok' → nil':mark':ok'
ok' :: nil':mark':ok' → nil':mark':ok'
top' :: nil':mark':ok' → top'
_hole_nil':mark':ok'1 :: nil':mark':ok'
_hole_top'2 :: top'
_gen_nil':mark':ok'3 :: Nat → nil':mark':ok'

Lemmas:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
app'(_gen_nil':mark':ok'3(+(1, _n2291)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2291)
from'(_gen_nil':mark':ok'3(+(1, _n5164))) → _*4, rt ∈ Ω(n5164)
s'(_gen_nil':mark':ok'3(+(1, _n6952))) → _*4, rt ∈ Ω(n6952)
zWadr'(_gen_nil':mark':ok'3(+(1, _n8864)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n8864)
prefix'(_gen_nil':mark':ok'3(+(1, _n12587))) → _*4, rt ∈ Ω(n12587)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_nil':mark':ok'3(+(1, _n5)), _gen_nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)