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(true, x, y) → f(gt(x, y), trunc(x), s(y))
trunc(0) → 0
trunc(s(0)) → 0
trunc(s(s(x))) → s(s(trunc(x)))
gt(0, v) → false
gt(s(u), 0) → true
gt(s(u), s(v)) → gt(u, v)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)