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(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(b) → a__b
mark(a) → a
a__f(X1, X2, X3) → f(X1, X2, X3)
a__b → b

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(X1, X2, X3)) →⁺ a__f(X1, mark(X2), X3)
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [X2 / f(X1, X2, X3)].
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(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(b) → a__b
mark(a) → a
a__f(X1, X2, X3) → f(X1, X2, X3)
a__b → b

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
a__f(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(b) → a__b
mark(a) → a
a__f(X1, X2, X3) → f(X1, X2, X3)
a__b → b

Types:
a__f :: a:b:f → a:b:f → a:b:f → a:b:f
a :: a:b:f
a__b :: a:b:f
b :: a:b:f
mark :: a:b:f → a:b:f
f :: a:b:f → a:b:f → a:b:f → a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat → a:b:f

(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(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(b) → a__b
mark(a) → a
a__f(X1, X2, X3) → f(X1, X2, X3)
a__b → b

Types:
a__f :: a:b:f → a:b:f → a:b:f → a:b:f
a :: a:b:f
a__b :: a:b:f
b :: a:b:f
mark :: a:b:f → a:b:f
f :: a:b:f → a:b:f → a:b:f → a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat → a:b:f

Generator Equations:
gen_a:b:f2_0(0) ⇔ a
gen_a:b:f2_0(+(x, 1)) ⇔ f(a, gen_a:b:f2_0(x), a)

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(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(b) → a__b
mark(a) → a
a__f(X1, X2, X3) → f(X1, X2, X3)
a__b → b

Types:
a__f :: a:b:f → a:b:f → a:b:f → a:b:f
a :: a:b:f
a__b :: a:b:f
b :: a:b:f
mark :: a:b:f → a:b:f
f :: a:b:f → a:b:f → a:b:f → a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat → a:b:f

Generator Equations:
gen_a:b:f2_0(0) ⇔ a
gen_a:b:f2_0(+(x, 1)) ⇔ f(a, gen_a:b:f2_0(x), a)

The following defined symbols remain to be analysed:
mark