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:

cond(true, x, y) → cond(and(gr(x, 0), gr(y, 0)), p(x), p(y))
and(true, true) → true
and(x, false) → false
and(false, x) → false
gr(0, 0) → false
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
cond(true, s(x15_1), s(x168_1)) →⁺ cond(true, x15_1, x168_1)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x15_1 / s(x15_1), x168_1 / s(x168_1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)