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(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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(X1), X2) →⁺ mark(f(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
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(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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, f, g, proper, top

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

(8) Obligation:

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

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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

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

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

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

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, proper, top

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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
g(gen_mark:ok3_0(+(1, n674_0))) → *4_0, rt ∈ Ω(n674₀)

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

Induction Step:
g(gen_mark:ok3_0(+(1, +(n674_0, 1)))) →_R^Ω(1)
mark(g(gen_mark:ok3_0(+(1, n674_0)))) →_IH
mark(*4_0)

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
g(gen_mark:ok3_0(+(1, n674_0))) → *4_0, rt ∈ Ω(n674₀)

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

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
g(gen_mark:ok3_0(+(1, n674_0))) → *4_0, rt ∈ Ω(n674₀)

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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
g(gen_mark:ok3_0(+(1, n674_0))) → *4_0, rt ∈ Ω(n674₀)

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

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

TRS:
Rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
g(gen_mark:ok3_0(+(1, n674_0))) → *4_0, rt ∈ Ω(n674₀)

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.

(21) LowerBoundsProof (EQUIVALENT transformation)

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

(22) BOUNDS(n^1, INF)

(23) Obligation:

TRS:
Rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
g(gen_mark:ok3_0(+(1, n674_0))) → *4_0, rt ∈ Ω(n674₀)

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.

(24) LowerBoundsProof (EQUIVALENT transformation)

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

(25) BOUNDS(n^1, INF)

(26) Obligation:

TRS:
Rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

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, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(28) BOUNDS(n^1, INF)