### (0) Obligation:

Runtime Complexity TRS:

The TRS R consists of the following rules:

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))

f(b(x), b(y)) → b(f(x, y))

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, 2]

transitions:

b0(0) → 0

a0(0) → 1

f0(0, 0) → 2

f1(0, 0) → 3

b1(3) → 2

b1(3) → 3

### (2) BOUNDS(O(1), O(n^1))