### (0) Obligation:

The Runtime Complexity (full) of the given

*CpxTRS* could be proven to be

BOUNDS(1, n^1).

The TRS R consists of the following rules:

f(x, y) → g(x, y)

g(h(x), y) → h(f(x, y))

g(h(x), y) → h(g(x, y))

Rewrite Strategy: FULL

### (1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

### (2) Obligation:

The Runtime Complexity (innermost) of the given

*CpxTRS* could be proven to be

BOUNDS(1, n^1).

The TRS R consists of the following rules:

f(x, y) → g(x, y)

g(h(x), y) → h(f(x, y))

g(h(x), y) → h(g(x, y))

Rewrite Strategy: INNERMOST

### (3) 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:

h0(0) → 0

f0(0, 0) → 1

g0(0, 0) → 2

g1(0, 0) → 1

f1(0, 0) → 3

h1(3) → 2

g1(0, 0) → 4

h1(4) → 2

g2(0, 0) → 3

h1(3) → 1

h1(3) → 4

h1(4) → 1

h1(4) → 4

h1(3) → 3

h1(4) → 3

### (4) BOUNDS(1, n^1)