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(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(mark(X)) →⁺ mark(f(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, c, f, g, d, h, proper, top

They will be analysed ascendingly in the following order:
c < active
f < active
g < active
d < active
h < active
active < top
c < proper
f < proper
g < proper
d < proper
h < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

The following defined symbols remain to be analysed:
c, active, f, g, d, h, proper, top

They will be analysed ascendingly in the following order:
c < active
f < active
g < active
d < active
h < active
active < top
c < proper
f < proper
g < proper
d < proper
h < proper
proper < top

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

The following defined symbols remain to be analysed:
f, active, g, d, h, proper, top

They will be analysed ascendingly in the following order:
f < active
g < active
d < active
h < active
active < top
f < proper
g < proper
d < proper
h < proper
proper < top

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)

Induction Base:
f(gen_mark:ok3_0(+(1, 0)))

Induction Step:
f(gen_mark:ok3_0(+(1, +(n9_0, 1)))) →_R^Ω(1)
mark(f(gen_mark:ok3_0(+(1, n9_0)))) →_IH
mark(*4_0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

The following defined symbols remain to be analysed:
g, active, d, h, proper, top

They will be analysed ascendingly in the following order:
g < active
d < active
h < active
active < top
g < proper
d < proper
h < proper
proper < top

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

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

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

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

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

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

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
h(gen_mark:ok3_0(+(1, n347_0))) → *4_0, rt ∈ Ω(n347₀)

Induction Base:
h(gen_mark:ok3_0(+(1, 0)))

Induction Step:
h(gen_mark:ok3_0(+(1, +(n347_0, 1)))) →_R^Ω(1)
mark(h(gen_mark:ok3_0(+(1, n347_0)))) →_IH
mark(*4_0)

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

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)
h(gen_mark:ok3_0(+(1, n347_0))) → *4_0, rt ∈ Ω(n347₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_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

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)
h(gen_mark:ok3_0(+(1, n347_0))) → *4_0, rt ∈ Ω(n347₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_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

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(24) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)
h(gen_mark:ok3_0(+(1, n347_0))) → *4_0, rt ∈ Ω(n347₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

The following defined symbols remain to be analysed:
top

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

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

(26) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)
h(gen_mark:ok3_0(+(1, n347_0))) → *4_0, rt ∈ Ω(n347₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)

(28) BOUNDS(n^1, INF)

(29) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)
h(gen_mark:ok3_0(+(1, n347_0))) → *4_0, rt ∈ Ω(n347₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

TRS:
Rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok

Lemmas:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)

Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
f(gen_mark:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n9₀)

(34) BOUNDS(n^1, INF)