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:

f(x, h(y)) → h(f(f(h(a), y), x))

Rewrite Strategy: INNERMOST

(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, h(y)) → h(f(f(h(a), y), x))

S is empty.
Rewrite Strategy: INNERMOST

(3) DecreasingLoopProof (EQUIVALENT transformation)

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

(4) BOUNDS(n^1, INF)