### (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:

h(f(x, y)) → f(f(a, h(h(y))), x)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, 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:

h(f(x, y)) → f(f(a, h(h(y))), x)

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]

transitions:

f0(0, 0) → 0

a0() → 0

h0(0) → 1

a1() → 3

h1(0) → 5

h1(5) → 4

f1(3, 4) → 2

f1(2, 0) → 1

f1(2, 0) → 5

a2() → 7

h2(0) → 9

h2(9) → 8

f2(7, 8) → 6

f2(6, 2) → 4

f1(2, 0) → 9

f2(6, 2) → 8

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