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:

plus_x#1(0, x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
plus_x#1(S(x12), x14) →⁺ S(plus_x#1(x12, x14))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x12 / S(x12)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)