### (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(s(x)) → s(s(f(p(s(x)))))

f(0) → 0

p(s(x)) → 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:

f(s(x)) → s(s(f(p(s(x)))))

f(0) → 0

p(s(x)) → x

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 2.

The certificate found is represented by the following graph.

Start state: 3

Accept states: [4]

Transitions:

3→4[f_1|0, p_1|0, 0|1, s_1|1]

3→5[s_1|1]

4→4[s_1|0, 0|0]

5→6[s_1|1]

6→7[f_1|1]

6→9[s_1|2]

6→4[0|2]

7→8[p_1|1]

7→4[s_1|1, 0|1]

8→4[s_1|1]

9→10[s_1|2]

10→11[f_1|2]

10→9[s_1|2]

10→4[0|2]

11→12[p_1|2]

11→4[s_1|1, 0|1]

12→4[s_1|2]

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