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

Types:
active :: mark:b:c:ok → mark:b:c:ok
f :: mark:b:c:ok → mark:b:c:ok → mark:b:c:ok → mark:b:c:ok
g :: mark:b:c:ok → mark:b:c:ok
mark :: mark:b:c:ok → mark:b:c:ok
b :: mark:b:c:ok
c :: mark:b:c:ok
proper :: mark:b:c:ok → mark:b:c:ok
ok :: mark:b:c:ok → mark:b:c:ok
top :: mark:b:c:ok → top
hole_mark:b:c:ok1_0 :: mark:b:c:ok
hole_top2_0 :: top
gen_mark:b:c:ok3_0 :: Nat → mark:b:c: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(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Generator Equations:
gen_mark:b:c:ok3_0(0) ⇔ b
gen_mark:b:c:ok3_0(+(x, 1)) ⇔ mark(gen_mark:b:c: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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

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

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

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

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

Proved the following rewrite lemma:
g(gen_mark:b:c:ok3_0(+(1, n17_0))) → *4_0, rt ∈ Ω(n17₀)

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

Induction Step:
g(gen_mark:b:c:ok3_0(+(1, +(n17_0, 1)))) →_R^Ω(1)
mark(g(gen_mark:b:c:ok3_0(+(1, n17_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(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
active(g(X)) → g(active(X))
g(mark(X)) → mark(g(X))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(g(X)) → g(proper(X))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

Lemmas:
g(gen_mark:b:c:ok3_0(+(1, n17_0))) → *4_0, rt ∈ Ω(n17₀)

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

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

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

(15) Obligation:

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

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

Lemmas:
g(gen_mark:b:c:ok3_0(+(1, n17_0))) → *4_0, rt ∈ Ω(n17₀)

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

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

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

(17) Obligation:

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

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

Lemmas:
g(gen_mark:b:c:ok3_0(+(1, n17_0))) → *4_0, rt ∈ Ω(n17₀)

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

The following defined symbols remain to be analysed:
top

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

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

(19) Obligation:

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

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

Lemmas:
g(gen_mark:b:c:ok3_0(+(1, n17_0))) → *4_0, rt ∈ Ω(n17₀)

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

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_mark:b:c:ok3_0(+(1, n17_0))) → *4_0, rt ∈ Ω(n17₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

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

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

Lemmas:
g(gen_mark:b:c:ok3_0(+(1, n17_0))) → *4_0, rt ∈ Ω(n17₀)

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

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_mark:b:c:ok3_0(+(1, n17_0))) → *4_0, rt ∈ Ω(n17₀)

(24) BOUNDS(n^1, INF)