The Runtime Complexity (full) 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:

+(0, y) → y
+(s(x), 0) → s(x)
+(s(x), s(y)) → s(+(s(x), +(y, 0)))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
+(s(x), s(s(x14_1))) →⁺ s(+(s(x), s(x14_1)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x14_1 / s(x14_1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)