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

(0) Obligation:

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

power(x', Cons(x, xs)) → mult(x', power(x', xs))
mult(x', Cons(x, xs)) → add0(x', mult(x', xs))
add0(x', Cons(x, xs)) → Cons(Cons(Nil, Nil), add0(x', xs))
power(x, Nil) → Cons(Nil, Nil)
mult(x, Nil) → Nil
add0(x, Nil) → x
goal(x, y) → power(x, 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', Cons(x, xs)) →⁺ mult(x', power(x', xs))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs / Cons(x, xs)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)