(0) 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(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(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]
1→3[s_1|1]
1→4[f_1|1]
1→7[s_1|2]
2→2[0|0, s_1|0]
3→2[0|1]
4→5[f_1|1]
4→6[s_1|1]
4→4[f_1|1]
4→7[s_1|2]
5→2[s_1|1]
6→2[0|1]
7→2[0|2]