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(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(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(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(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(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

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

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

(6) Obligation:

TRS:
Rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))

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

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

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

Proved the following rewrite lemma:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
g(gen_mark:a:b:ok3_0(+(1, 0)), gen_mark:a:b:ok3_0(b))

Induction Step:
g(gen_mark:a:b:ok3_0(+(1, +(n5_0, 1))), gen_mark:a:b:ok3_0(b)) →_R^Ω(1)
mark(g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b: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(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))

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

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

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

Proved the following rewrite lemma:
f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n782₀)

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

Induction Step:
f(gen_mark:a:b:ok3_0(+(1, +(n782_0, 1))), gen_mark:a:b:ok3_0(b)) →_R^Ω(1)
mark(f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b))) →_IH
mark(*4_0)

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

(11) Complex Obligation (BEST)

(12) Obligation:

TRS:
Rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n782₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b: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

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

Proved the following rewrite lemma:
h(gen_mark:a:b:ok3_0(+(1, n1863_0))) → *4_0, rt ∈ Ω(n1863₀)

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

Induction Step:
h(gen_mark:a:b:ok3_0(+(1, +(n1863_0, 1)))) →_R^Ω(1)
mark(h(gen_mark:a:b:ok3_0(+(1, n1863_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(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n782₀)
h(gen_mark:a:b:ok3_0(+(1, n1863_0))) → *4_0, rt ∈ Ω(n1863₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b: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

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

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

(17) Obligation:

TRS:
Rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n782₀)
h(gen_mark:a:b:ok3_0(+(1, n1863_0))) → *4_0, rt ∈ Ω(n1863₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))

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

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

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n782₀)
h(gen_mark:a:b:ok3_0(+(1, n1863_0))) → *4_0, rt ∈ Ω(n1863₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))

The following defined symbols remain to be analysed:
top

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(21) Obligation:

TRS:
Rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n782₀)
h(gen_mark:a:b:ok3_0(+(1, n1863_0))) → *4_0, rt ∈ Ω(n1863₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

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

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n782₀)
h(gen_mark:a:b:ok3_0(+(1, n1863_0))) → *4_0, rt ∈ Ω(n1863₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

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

(26) BOUNDS(n^1, INF)

(27) Obligation:

TRS:
Rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
f(gen_mark:a:b:ok3_0(+(1, n782_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n782₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

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

(29) BOUNDS(n^1, INF)

(30) Obligation:

TRS:
Rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:a:b:ok3_0(+(1, n5_0)), gen_mark:a:b:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

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

(32) BOUNDS(n^1, INF)