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:

a__f(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(mark(X))
a__f(X) → f(X)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

a__f(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(mark(X))
a__f(X) → f(X)

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
g/0

(6) Obligation:

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

a__f(X) → g
mark(f(X)) → a__f(mark(X))
mark(g) → g
mark(h(X)) → h(mark(X))
a__f(X) → f(X)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
a__f(X) → g
mark(f(X)) → a__f(mark(X))
mark(g) → g
mark(h(X)) → h(mark(X))
a__f(X) → f(X)

Types:
a__f :: g:f:h → g:f:h
g :: g:f:h
mark :: g:f:h → g:f:h
f :: g:f:h → g:f:h
h :: g:f:h → g:f:h
hole_g:f:h1_0 :: g:f:h
gen_g:f:h2_0 :: Nat → g:f:h

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mark

(10) Obligation:

TRS:
Rules:
a__f(X) → g
mark(f(X)) → a__f(mark(X))
mark(g) → g
mark(h(X)) → h(mark(X))
a__f(X) → f(X)

Types:
a__f :: g:f:h → g:f:h
g :: g:f:h
mark :: g:f:h → g:f:h
f :: g:f:h → g:f:h
h :: g:f:h → g:f:h
hole_g:f:h1_0 :: g:f:h
gen_g:f:h2_0 :: Nat → g:f:h

Generator Equations:
gen_g:f:h2_0(0) ⇔ g
gen_g:f:h2_0(+(x, 1)) ⇔ f(gen_g:f:h2_0(x))

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:
mark(gen_g:f:h2_0(n4_0)) → gen_g:f:h2_0(0), rt ∈ Ω(1 + n4₀)

Induction Base:
mark(gen_g:f:h2_0(0)) →_R^Ω(1)
g

Induction Step:
mark(gen_g:f:h2_0(+(n4_0, 1))) →_R^Ω(1)
a__f(mark(gen_g:f:h2_0(n4_0))) →_IH
a__f(gen_g:f:h2_0(0)) →_R^Ω(1)
g

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
a__f(X) → g
mark(f(X)) → a__f(mark(X))
mark(g) → g
mark(h(X)) → h(mark(X))
a__f(X) → f(X)

Types:
a__f :: g:f:h → g:f:h
g :: g:f:h
mark :: g:f:h → g:f:h
f :: g:f:h → g:f:h
h :: g:f:h → g:f:h
hole_g:f:h1_0 :: g:f:h
gen_g:f:h2_0 :: Nat → g:f:h

Lemmas:
mark(gen_g:f:h2_0(n4_0)) → gen_g:f:h2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_g:f:h2_0(0) ⇔ g
gen_g:f:h2_0(+(x, 1)) ⇔ f(gen_g:f:h2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_g:f:h2_0(n4_0)) → gen_g:f:h2_0(0), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
a__f(X) → g
mark(f(X)) → a__f(mark(X))
mark(g) → g
mark(h(X)) → h(mark(X))
a__f(X) → f(X)

Types:
a__f :: g:f:h → g:f:h
g :: g:f:h
mark :: g:f:h → g:f:h
f :: g:f:h → g:f:h
h :: g:f:h → g:f:h
hole_g:f:h1_0 :: g:f:h
gen_g:f:h2_0 :: Nat → g:f:h

Lemmas:
mark(gen_g:f:h2_0(n4_0)) → gen_g:f:h2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_g:f:h2_0(0) ⇔ g
gen_g:f:h2_0(+(x, 1)) ⇔ f(gen_g:f:h2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_g:f:h2_0(n4_0)) → gen_g:f:h2_0(0), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)