### (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]