The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

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

f(0) → 1
f(s(x)) → g(f(x))
g(x) → +(x, s(x))
f(s(x)) → +(f(x), s(f(x)))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
f(s(s(x6_2))) →⁺ g(+(f(x6_2), s(f(x6_2))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [x6_2 / s(s(x6_2))].
The result substitution is [ ].

The rewrite sequence
f(s(s(x6_2))) →⁺ g(+(f(x6_2), s(f(x6_2))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].
The pumping substitution is [x6_2 / s(s(x6_2))].
The result substitution is [ ].

(2) BOUNDS(2^n, INF)