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:

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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(x, c(y)) →⁺ f(x, s(f(y, y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
The pumping substitution is [y / c(y)].
The result substitution is [x / y].

(2) BOUNDS(n^1, INF)