The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

Runtime Complexity TRS:
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

(1) 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: 1
Accept states: [2]
Transitions:
1→2[f_1|0, p_1|0, 0|1, s_1|1]
1→3[s_1|1]
2→2[s_1|0, 0|0]
3→4[s_1|1]
4→5[f_1|1]
4→7[s_1|2]
4→2[0|2]
5→6[p_1|1]
5→2[s_1|1, 0|1]
6→2[s_1|1]
7→8[s_1|2]
8→9[f_1|2]
8→7[s_1|2]
8→2[0|2]
9→10[p_1|2]
9→2[s_1|1, 0|1]
10→2[s_1|2]

(2) BOUNDS(1, n^1)