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:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
inc(0) → 0
inc(s(x)) → s(inc(x))
zero(0) → true
zero(s(x)) → false
p(0) → 0
p(s(x)) → x
bits(x) → bitIter(x, 0)
bitIter(x, y) → if(zero(x), x, inc(y))
if(true, x, y) → p(y)
if(false, x, y) → bitIter(half(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
half(s(s(x))) →⁺ s(half(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(s(x))].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)