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:

g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)