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:

minus(x, y) → cond(ge(x, s(y)), x, y)
cond(false, x, y) → 0
cond(true, x, y) → s(minus(x, s(y)))
ge(u, 0) → true
ge(0, s(v)) → false
ge(s(u), s(v)) → ge(u, v)

Rewrite Strategy: FULL

(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:

minus(x, y) → cond(ge(x, s(y)), x, y)
cond(false, x, y) → 0
cond(true, x, y) → s(minus(x, s(y)))
ge(u, 0) → true
ge(0, s(v)) → false
ge(s(u), s(v)) → ge(u, v)

S is empty.
Rewrite Strategy: FULL

(3) DecreasingLoopProof (EQUIVALENT transformation)

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

(4) BOUNDS(n^1, INF)