The TRS does not nest defined symbols. Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: a(a(f(x, y))) → f(a(b(a(b(a(x))))), a(b(a(b(a(y)))))) f(a(x), a(y)) → a(f(x, y))
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
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 :  transitions: b0(0) → 0 f0(0, 0) → 1 f1(0, 0) → 2 b1(2) → 1 b1(2) → 2