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

(0) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

power(x', S(x)) → mult(x', power(x', x))
mult(x', S(x)) → add0(x', mult(x', x))
add0(x', S(x)) → +(S(0), add0(x', x))
power(x, 0) → S(0)
mult(x, 0) → 0
add0(x, 0) → x

The (relative) TRS S consists of the following rules:

+(x, S(0)) → S(x)
+(S(0), y) → S(y)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
power(x', S(x)) →⁺ mult(x', power(x', x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / S(x)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)