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(0, x) → 0
minus(s(x), 0) → s(x)
minus(s(x), s(y)) → minus(x, y)
mod(x, 0) → 0
mod(x, s(y)) → if(lt(x, s(y)), x, s(y))
if(true, x, y) → x
if(false, x, y) → mod(minus(x, y), y)
gcd(x, 0) → x
gcd(0, s(y)) → s(y)
gcd(s(x), s(y)) → gcd(mod(s(x), s(y)), mod(s(y), s(x)))
lt(x, 0) → false
lt(0, s(x)) → true
lt(s(x), s(y)) → lt(x, y)

Rewrite Strategy: FULL

(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)) →⁺ minus(x, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), y / s(y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)