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:

perfectp(0) → false
perfectp(s(x)) → f(x, s(0), s(x), s(x))
f(0, y, 0, u) → true
f(0, y, s(z), u) → false
f(s(x), 0, z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(s(s(x368661_1)), 0, z, 0) →⁺ f(x368661_1, 0, minus(minus(z, s(s(x368661_1))), s(x368661_1)), 0)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x368661_1 / s(s(x368661_1))].
The result substitution is [z / minus(minus(z, s(s(x368661_1))), s(x368661_1))].

(2) BOUNDS(n^1, INF)