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

Types:
a__f :: a:f:g → a:f:g
f :: a:f:g → a:f:g
a :: a:f:g
g :: a:f:g
mark :: a:f:g → a:f:g
hole_a:f:g1_0 :: a:f:g
gen_a:f:g2_0 :: Nat → a:f:g

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

Heuristically decided to analyse the following defined symbols:
a__f, mark

They will be analysed ascendingly in the following order:
a__f < mark

(8) Obligation:

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

Types:
a__f :: a:f:g → a:f:g
f :: a:f:g → a:f:g
a :: a:f:g
g :: a:f:g
mark :: a:f:g → a:f:g
hole_a:f:g1_0 :: a:f:g
gen_a:f:g2_0 :: Nat → a:f:g

Generator Equations:
gen_a:f:g2_0(0) ⇔ a
gen_a:f:g2_0(+(x, 1)) ⇔ f(gen_a:f:g2_0(x))

The following defined symbols remain to be analysed:
a__f, mark

They will be analysed ascendingly in the following order:
a__f < mark

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

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

Types:
a__f :: a:f:g → a:f:g
f :: a:f:g → a:f:g
a :: a:f:g
g :: a:f:g
mark :: a:f:g → a:f:g
hole_a:f:g1_0 :: a:f:g
gen_a:f:g2_0 :: Nat → a:f:g

Generator Equations:
gen_a:f:g2_0(0) ⇔ a
gen_a:f:g2_0(+(x, 1)) ⇔ f(gen_a:f:g2_0(x))

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:
mark(gen_a:f:g2_0(+(1, n14_0))) → *3_0, rt ∈ Ω(n14₀)

Induction Base:
mark(gen_a:f:g2_0(+(1, 0)))

Induction Step:
mark(gen_a:f:g2_0(+(1, +(n14_0, 1)))) →_R^Ω(1)
a__f(mark(gen_a:f:g2_0(+(1, n14_0)))) →_IH
a__f(*3_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:
a__f(f(a)) → a__f(g)
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(g) → g
a__f(X) → f(X)

Types:
a__f :: a:f:g → a:f:g
f :: a:f:g → a:f:g
a :: a:f:g
g :: a:f:g
mark :: a:f:g → a:f:g
hole_a:f:g1_0 :: a:f:g
gen_a:f:g2_0 :: Nat → a:f:g

Lemmas:
mark(gen_a:f:g2_0(+(1, n14_0))) → *3_0, rt ∈ Ω(n14₀)

Generator Equations:
gen_a:f:g2_0(0) ⇔ a
gen_a:f:g2_0(+(x, 1)) ⇔ f(gen_a:f:g2_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_a:f:g2_0(+(1, n14_0))) → *3_0, rt ∈ Ω(n14₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
a__f :: a:f:g → a:f:g
f :: a:f:g → a:f:g
a :: a:f:g
g :: a:f:g
mark :: a:f:g → a:f:g
hole_a:f:g1_0 :: a:f:g
gen_a:f:g2_0 :: Nat → a:f:g

Lemmas:
mark(gen_a:f:g2_0(+(1, n14_0))) → *3_0, rt ∈ Ω(n14₀)

Generator Equations:
gen_a:f:g2_0(0) ⇔ a
gen_a:f:g2_0(+(x, 1)) ⇔ f(gen_a:f:g2_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_a:f:g2_0(+(1, n14_0))) → *3_0, rt ∈ Ω(n14₀)

(18) BOUNDS(n^1, INF)