The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

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

f(S(x), x2) → f(x2, x)
f(0, x2) → 0

Rewrite Strategy: INNERMOST

(1) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
S0(0) → 0
00() → 0
f0(0, 0) → 1
f1(0, 0) → 1
01() → 1

(2) BOUNDS(1, n^1)