The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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: FULL

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative 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))

S is empty.
Rewrite Strategy: FULL

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

(5) OrderProof (LOWER BOUND(ID) transformation)

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

(6) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
cons(gen_nil:mark:ok3_0(+(1, 0)), gen_nil:mark:ok3_0(b))

Induction Step:
cons(gen_nil:mark:ok3_0(+(1, +(n5_0, 1))), gen_nil:mark:ok3_0(b)) →_R^Ω(1)
mark(cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)

Induction Base:
app(gen_nil:mark:ok3_0(+(1, 0)), gen_nil:mark:ok3_0(b))

Induction Step:
app(gen_nil:mark:ok3_0(+(1, +(n998_0, 1))), gen_nil:mark:ok3_0(b)) →_R^Ω(1)
mark(app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(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

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)

Induction Base:
from(gen_nil:mark:ok3_0(+(1, 0)))

Induction Step:
from(gen_nil:mark:ok3_0(+(1, +(n2494_0, 1)))) →_R^Ω(1)
mark(from(gen_nil:mark:ok3_0(+(1, n2494_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(14) Complex Obligation (BEST)

(15) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(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

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)

Induction Base:
s(gen_nil:mark:ok3_0(+(1, 0)))

Induction Step:
s(gen_nil:mark:ok3_0(+(1, +(n3175_0, 1)))) →_R^Ω(1)
mark(s(gen_nil:mark:ok3_0(+(1, n3175_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(17) Complex Obligation (BEST)

(18) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(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

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n3957₀)

Induction Base:
zWadr(gen_nil:mark:ok3_0(+(1, 0)), gen_nil:mark:ok3_0(b))

Induction Step:
zWadr(gen_nil:mark:ok3_0(+(1, +(n3957_0, 1))), gen_nil:mark:ok3_0(b)) →_R^Ω(1)
mark(zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(20) Complex Obligation (BEST)

(21) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)
zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n3957₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(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

(22) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
prefix(gen_nil:mark:ok3_0(+(1, n6169_0))) → *4_0, rt ∈ Ω(n6169₀)

Induction Base:
prefix(gen_nil:mark:ok3_0(+(1, 0)))

Induction Step:
prefix(gen_nil:mark:ok3_0(+(1, +(n6169_0, 1)))) →_R^Ω(1)
mark(prefix(gen_nil:mark:ok3_0(+(1, n6169_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(23) Complex Obligation (BEST)

(24) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)
zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n3957₀)
prefix(gen_nil:mark:ok3_0(+(1, n6169_0))) → *4_0, rt ∈ Ω(n6169₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(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

(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(26) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)
zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n3957₀)
prefix(gen_nil:mark:ok3_0(+(1, n6169_0))) → *4_0, rt ∈ Ω(n6169₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

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

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

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(28) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)
zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n3957₀)
prefix(gen_nil:mark:ok3_0(+(1, n6169_0))) → *4_0, rt ∈ Ω(n6169₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

The following defined symbols remain to be analysed:
top

(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(30) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)
zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n3957₀)
prefix(gen_nil:mark:ok3_0(+(1, n6169_0))) → *4_0, rt ∈ Ω(n6169₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(32) BOUNDS(n^1, INF)

(33) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)
zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n3957₀)
prefix(gen_nil:mark:ok3_0(+(1, n6169_0))) → *4_0, rt ∈ Ω(n6169₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(35) BOUNDS(n^1, INF)

(36) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)
zWadr(gen_nil:mark:ok3_0(+(1, n3957_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n3957₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(38) BOUNDS(n^1, INF)

(39) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)
s(gen_nil:mark:ok3_0(+(1, n3175_0))) → *4_0, rt ∈ Ω(n3175₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

No more defined symbols left to analyse.

(40) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(41) BOUNDS(n^1, INF)

(42) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)
from(gen_nil:mark:ok3_0(+(1, n2494_0))) → *4_0, rt ∈ Ω(n2494₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

No more defined symbols left to analyse.

(43) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(44) BOUNDS(n^1, INF)

(45) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
app(gen_nil:mark:ok3_0(+(1, n998_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n998₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

No more defined symbols left to analyse.

(46) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(47) BOUNDS(n^1, INF)

(48) Obligation:

TRS:
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:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok

Lemmas:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))

No more defined symbols left to analyse.

(49) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_nil:mark:ok3_0(+(1, n5_0)), gen_nil:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(50) BOUNDS(n^1, INF)