The Runtime Complexity (innermost) 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: INNERMOST

(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)