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:

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(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 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
f0(0) → 0
g0(0, 0) → 1
h0(0, 0) → 2
h1(0, 0) → 3
f1(3) → 1
f1(0) → 4
g1(0, 4) → 2
h1(0, 4) → 3
f1(3) → 2
f2(0) → 5
g2(0, 5) → 3
f2(4) → 5
h1(0, 5) → 3
f1(3) → 3
f2(5) → 5

(2) BOUNDS(1, n^1)