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:

dbl(S(0), S(0)) → S(S(S(S(0))))
save(S(x)) → dbl(0, save(x))
save(0) → 0
dbl(0, y) → y

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
save(S(x)) →⁺ dbl(0, save(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)