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(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
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(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
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(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

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

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

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

(6) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

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

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

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

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

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

Induction Base:
f(gen_a:b:mark:c:ok3_0(a), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(+(1, 0)))

Induction Step:
f(gen_a:b:mark:c:ok3_0(a), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
mark(f(gen_a:b:mark:c:ok3_0(a), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(+(1, n5_0)))) →_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(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(a), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(a), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

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

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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(a), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

The following defined symbols remain to be analysed:
top

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

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

(15) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(a), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

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

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: a:b:mark:c:ok → a:b:mark:c:ok
f :: a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok → a:b:mark:c:ok
a :: a:b:mark:c:ok
b :: a:b:mark:c:ok
mark :: a:b:mark:c:ok → a:b:mark:c:ok
c :: a:b:mark:c:ok
proper :: a:b:mark:c:ok → a:b:mark:c:ok
ok :: a:b:mark:c:ok → a:b:mark:c:ok
top :: a:b:mark:c:ok → top
hole_a:b:mark:c:ok1_0 :: a:b:mark:c:ok
hole_top2_0 :: top
gen_a:b:mark:c:ok3_0 :: Nat → a:b:mark:c:ok

Lemmas:
f(gen_a:b:mark:c:ok3_0(a), gen_a:b:mark:c:ok3_0(b), gen_a:b:mark:c:ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

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

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

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

(20) BOUNDS(n^1, INF)