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:

add0(S(x), x2) → +(S(0), add0(x2, x))
add0(0, x2) → x2

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
add0(S(x), S(x11_1)) →⁺ +(S(0), +(S(0), add0(x, x11_1)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [x / S(x), x11_1 / S(x11_1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)