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) 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:

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

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
g/0

(4) 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

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

Infered types.

(6) 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

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
mark

(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

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

(9) 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).

(10) Complex Obligation (BEST)

(11) 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.

(12) 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)

(13) BOUNDS(n^1, INF)

(14) 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.

(15) 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)

(16) BOUNDS(n^1, INF)