(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:
odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0) → 0
even(0) → S(0)
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 1.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[odd_1|0, even_1|0, even_1|1, 0|1, odd_1|1]
1→3[S_1|1]
2→2[S_1|0, 0|0]
3→2[0|1]