### (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:

g(s(x)) → f(x)

f(0) → s(0)

f(s(x)) → s(s(g(x)))

g(0) → 0

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:

g(s(x)) → f(x)

f(0) → s(0)

f(s(x)) → s(s(g(x)))

g(0) → 0

Rewrite Strategy: INNERMOST

### (3) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.

The certificate found is represented by the following graph.

Start state: 3

Accept states: [4]

Transitions:

3→4[g_1|0, f_1|0, f_1|1, 0|1]

3→5[s_1|1]

3→6[s_1|1]

4→4[s_1|0, 0|0]

5→4[0|1]

6→7[s_1|1]

7→4[g_1|1, f_1|1, 0|1]

7→5[s_1|1]

7→6[s_1|1]

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