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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

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

Heuristically decided to analyse the following defined symbols:
active, cons, from, s, sel, minus, quot, zWquot, proper, top

They will be analysed ascendingly in the following order:
cons < active
from < active
s < active
sel < active
minus < active
quot < active
zWquot < active
active < top
cons < proper
from < proper
s < proper
sel < proper
minus < proper
quot < proper
zWquot < proper
proper < top

(6) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

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

The following defined symbols remain to be analysed:
cons, active, from, s, sel, minus, quot, zWquot, proper, top

They will be analysed ascendingly in the following order:
cons < active
from < active
s < active
sel < active
minus < active
quot < active
zWquot < active
active < top
cons < proper
from < proper
s < proper
sel < proper
minus < proper
quot < proper
zWquot < proper
proper < top

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

Proved the following rewrite lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
cons(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))

Induction Step:
cons(gen_mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil: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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

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

The following defined symbols remain to be analysed:
from, active, s, sel, minus, quot, zWquot, proper, top

They will be analysed ascendingly in the following order:
from < active
s < active
sel < active
minus < active
quot < active
zWquot < active
active < top
from < proper
s < proper
sel < proper
minus < proper
quot < proper
zWquot < proper
proper < top

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

Proved the following rewrite lemma:
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)

Induction Base:
from(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
from(gen_mark:0':nil:ok3_0(+(1, +(n1142_0, 1)))) →_R^Ω(1)
mark(from(gen_mark:0':nil:ok3_0(+(1, n1142_0)))) →_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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)

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

The following defined symbols remain to be analysed:
s, active, sel, minus, quot, zWquot, proper, top

They will be analysed ascendingly in the following order:
s < active
sel < active
minus < active
quot < active
zWquot < active
active < top
s < proper
sel < proper
minus < proper
quot < proper
zWquot < proper
proper < top

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

Proved the following rewrite lemma:
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)

Induction Base:
s(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
s(gen_mark:0':nil:ok3_0(+(1, +(n1709_0, 1)))) →_R^Ω(1)
mark(s(gen_mark:0':nil:ok3_0(+(1, n1709_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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)

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

The following defined symbols remain to be analysed:
sel, active, minus, quot, zWquot, proper, top

They will be analysed ascendingly in the following order:
sel < active
minus < active
quot < active
zWquot < active
active < top
sel < proper
minus < proper
quot < proper
zWquot < proper
proper < top

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

Proved the following rewrite lemma:
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)

Induction Base:
sel(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))

Induction Step:
sel(gen_mark:0':nil:ok3_0(+(1, +(n2377_0, 1))), gen_mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b))) →_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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)

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

The following defined symbols remain to be analysed:
minus, active, quot, zWquot, proper, top

They will be analysed ascendingly in the following order:
minus < active
quot < active
zWquot < active
active < top
minus < proper
quot < proper
zWquot < proper
proper < top

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

Proved the following rewrite lemma:
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)

Induction Base:
minus(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))

Induction Step:
minus(gen_mark:0':nil:ok3_0(+(1, +(n4429_0, 1))), gen_mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil: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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)

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

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

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

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

Proved the following rewrite lemma:
quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6785₀)

Induction Base:
quot(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))

Induction Step:
quot(gen_mark:0':nil:ok3_0(+(1, +(n6785_0, 1))), gen_mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b))) →_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(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)
quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6785₀)

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

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

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

(25) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
zWquot(gen_mark:0':nil:ok3_0(+(1, n9445_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9445₀)

Induction Base:
zWquot(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))

Induction Step:
zWquot(gen_mark:0':nil:ok3_0(+(1, +(n9445_0, 1))), gen_mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(zWquot(gen_mark:0':nil:ok3_0(+(1, n9445_0)), gen_mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

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

(26) Complex Obligation (BEST)

(27) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)
quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6785₀)
zWquot(gen_mark:0':nil:ok3_0(+(1, n9445_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9445₀)

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

(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(29) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)
quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6785₀)
zWquot(gen_mark:0':nil:ok3_0(+(1, n9445_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9445₀)

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

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

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

(30) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(31) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)
quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6785₀)
zWquot(gen_mark:0':nil:ok3_0(+(1, n9445_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9445₀)

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

The following defined symbols remain to be analysed:
top

(32) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(33) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)
quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6785₀)
zWquot(gen_mark:0':nil:ok3_0(+(1, n9445_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9445₀)

Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil: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_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(35) BOUNDS(n^1, INF)

(36) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)
quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6785₀)
zWquot(gen_mark:0':nil:ok3_0(+(1, n9445_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n9445₀)

Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil: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_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(38) BOUNDS(n^1, INF)

(39) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)
quot(gen_mark:0':nil:ok3_0(+(1, n6785_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6785₀)

Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil: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_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(41) BOUNDS(n^1, INF)

(42) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)
minus(gen_mark:0':nil:ok3_0(+(1, n4429_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n4429₀)

Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil: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_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(44) BOUNDS(n^1, INF)

(45) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)
sel(gen_mark:0':nil:ok3_0(+(1, n2377_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n2377₀)

Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil: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_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(47) BOUNDS(n^1, INF)

(48) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)
s(gen_mark:0':nil:ok3_0(+(1, n1709_0))) → *4_0, rt ∈ Ω(n1709₀)

Generator Equations:
gen_mark:0':nil:ok3_0(0) ⇔ 0'
gen_mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil: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_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(50) BOUNDS(n^1, INF)

(51) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
from(gen_mark:0':nil:ok3_0(+(1, n1142_0))) → *4_0, rt ∈ Ω(n1142₀)

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

No more defined symbols left to analyse.

(52) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(53) BOUNDS(n^1, INF)

(54) Obligation:

TRS:
Rules:
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0', cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0')) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0', s(Y))) → mark(0')
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
from :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
sel :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
minus :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
quot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
zWquot :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

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

No more defined symbols left to analyse.

(55) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(56) BOUNDS(n^1, INF)