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:

minus(X, 0) → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0, s(Y)) → 0
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)