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, a) → x
f(x, g(y)) → f(g(x), y)

Rewrite Strategy: FULL

(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) Obligation:

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

f(x, a) → x
f(x, g(y)) → f(g(x), y)

S is empty.
Rewrite Strategy: FULL

(3) DecreasingLoopProof (EQUIVALENT transformation)

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

(4) BOUNDS(n^1, INF)